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0; (b) g2(x) = 2"(2 + 14x11°), p > 0; (C) g3(X) := 2 + log(1 + IIxII).
In the last two examples, note that x H II x I I ° and x - log(1 + II x II) are subadditive functions on X. Proposition 1.8.13. For µ = [a, R, MI and for the submultiplicative function g which is continuous at zero, we have
f g(x)µ(dx) < oo if and only if fXII>Ig(x)M(dx) < oo. Proof. This follows from Propositions 1.8.5 and 1.8.12, the inequality (1.8.7), and the following inequalities: 0o
f g(x)e(m)(dx) 5 e--(x) g(0) + X
e(m)(A) >_ e-m(x)m(A).
k=I
k-1
k.
k
(fXg(x)m(dx))
,
1
Q.E.D.
Remark 1.8.14. Let
IDiog(X) = {µ E ID(X): fx log(1 + Ilxll)µ(dx) < oo). Then IDlog(X) is a semigroup and, for A E End(X), A(ID10g(X)) c IDiog(X). Note that if µ has Levy spectral measure M, then Aµ has Levy spectral measure AM.
37
INFINITELY DIVISIBLE PROBABILITIES
Theorem 1.8.15. Let H be a Hilbert space and let [an, Rn, Mn], [a, R, M] E ID(H). Then [an, Rn, Mn] - [a, R, MI if and only if the following hold.
(i) an - a in H. M outside of every neighborhood of zero. (ii) Mn (iii) For the S-operators, Tn, given by (Y, TTY)
f (x,y)2Mn(dx) + (Y,Rny), B,
we have lim inf
1i m lim sup (fB (x, Y)2Mn(dx) + (Y, RnY)
(a)
n-W
,
= (Y, RY )
for every y E H;
sup trace(7,)
_ 1, 1 < k < kn) of probabilities on Rd is shift weakly convergent to [a, R, M] E ID(l ) if and only if (a) outside of every neighborhood of zero,
k E 12n, k
k=1
lim inf
(b)
Im
CIO
I
limsup n_ao
for every y E
- M;
k f ( Y, x)'An,k(dx) = CY, RY)
k=1 B,
Rd.
We end this section with some facts about factors and supports of Gaussian measures. Theorem 1.8.18 (Cramer Theorem). Let y be a centered Gaussian measure on a Banach space X. Assume y = y1 * 72 for some y11 Y2 E 150(X). Then both y1 and 72 are Gaussian measures.
Theorem 1.8.19 (Skitovich Theorem). Let be a nondegenerate OBd-valued random variable. Then is Gaussian if and only if there exists a coordinate system for Rd, (x1, ... , xd), such that the projections
of
onto the x;-axes are independent, and there exist two more axes,
(y 1, y2), which are not contained in any proper hyperplane generated by {x1, ... , xd}, such that the projections of 6 onto the y1-axis and y2-axis are independent.
Finally, we wish to specify the support of a Gaussian measure. Let us recall from Section 1.7 that the support of a measure is the smallest closed subset having measure one. Theorem 1.8.20
(a) The support of a centered Gaussian measure on a Banach space X is a linear subspace of X.
39
THE SKOROHOD SPACE
(b) If H is a real separable Hilbert space and y is a centered Gaussian measure on H with covariance operator D, then
supp y = (ker D)1 .
Proof. We only prove (b) since
it
is needed later. Let x E
(supp y)1 . Then, for all z e H,
(z,Dx) = fH (z, y) (x,y)y(dy) = 0. Hence, (supp y)1 c ker D. Conversely, for x E ker D, we have
(x, Dx) =
IH
(x, y)2y(dy) = 0,
so (x, y) = 0 for all y e H y-a.s. Hence, y(y E H: (x, y) = 0) = 1 when x e ker D. Thus,
suppyc n (ker D)
.
Q.E.D.
Corollary 1.8.21. For a centered Gaussian measure y on Rd with covariance operator D, we have dim(supp y) = rank(D).
Note that from the above two facts, the support of a Gaussian measure is always a linear space (cf. the second part of Section 3.4). 1.9 THE SKOROHOD SPACE Let (Cl, .f, P) be a complete probability space, that is,
contains all
subsets of a set with P-measure zero. Let (S, ,I) be a measurable space and let T be an abstract set. A mapping f : Cl X T -> S is said to be a random function if, for any t E T, the mapping t): Cl --> S is an S-valued random variable. Furthermore, if T and S are topological spaces, then a random function is said to be separable (relative to open sets in T and closed sets in S) if there exist a dense countable subset To of T and a set N of the P-measure zero such that for any
40
PRELIMINARIES
open G in T and any closed F in S we have
all tEGnTO}\ (wESZ:e(t,w)EFfor all tEG) cN. Note that the first set is in .91' and, therefore, so is the second one. We
will see that in rather general situations we may assume that the random function is separable. More precisely, if we agree to call two random functions I and 2 on fl x T stochastically equivalent whenever t) = following theorem.
Theorem 1.9.1.
t)} = 1, for each t e T, then we have the Let S be a separable locally compact metric space
and let T be an arbitrary separable metric space. For any random function on fl x T with values in S, there exists a stochastically taking values in a certain compact equivalent separable function extension S of S.
If T is either the real line or a segment of the real line, the random function is said to be a random process, and T may be viewed as time. Furthermore, we say that the random process f(t), t r= [a, b], is
without discontinuities of the second kind on [a, b] if the sample functions 6(w, - ) have at each t E (a, b) with probability one left and
right limits and possess at the points a and b a right and left limit, respectively. Many classes of random processes have sample paths (functions) without discontinuities of the second kind. For our needs recall the following theorem.
Theorem 1.9.2. If C(t), t E [a, b], is a separable stochastically continuous process with independent increments and values in a linear normed space IE, then it has no discontinuities of the second kind.
Theorem 1.9.3. If fi(t), t E [a, b], is a stochastically continuous process without discontinuities of the second kind and with values in a metric space S, then there exists an equivalent process fi(t), t E [a, b], whose sample functions are continuous from the right with probability one.
Let D(S; [a, b]) be the set of all functions x defined on [a, b] with
values in a complete metric space S that are right-continuous on [a, b], have left limits x(t - ) on (a, b), and x(b) = x(b - ). From the
THE SKOROHOD SPACE
41
above discussion [existence of separable version; random functions without discontinuities of the second kind; existence of a version with sample functions in D(S; [a, b])], we see that without loss of generality we may usually assume that the random function from SZ X T into
S is a mapping n: SZ - D(S; T), when T is a closed interval in the real line. Moreover, using the coordinate mappings IIt: D(S; T) -> S given by IIt(x) := x(t), we have w H (II`rt)(co) := 77(t, w) is an Svalued random variable, that is, measurable mapping from (fl, F) into (S, ./). Remark 1.9.4. Let f(t ), t E [a, b], be a random process with values in an infinite-dimensional Banach space X. We cannot directly
apply Theorem 1.9.1 to claim that there is a version 6 with sample functions in D(X; [a, b]). However, such a version exists, if f has independent increments and is stochastically continuous. The discussion preceding Remark 1.9.4 suggests that it would be helpful to treat a random function as a measurable mapping from (c, f) into (D(S; [a, b]), -9) for some a-field -9. A natural requirement is that J1 should coincide with the Q-algebra generated by all cylindrical sets (x E D(S; [a, b]): x(t1) E A1,..., x(tR) E
where t, e [a, b] and Ai E .Q(S) for all j = 1, ... , n. Note that the coordinate mappings must be measurable. Also, let us note that such a measurable mapping .77 is a separable random function; as To one can take an arbitrary separable dense subset of [a, b]. Moreover, to be able to apply results from Sections 1.6 or 1.7, we should have a metric d on D(S; [a, b]) such that 3 would be the Q-field generated by the Borel sets with respect to the metric d and D(S; [a, b]) would be a
separable metric space. Sometimes it also needs to be a complete separable metric space (cf. Theorem 1.6.6). The metric that suits these purposes was defined by A. V. Skorohod and is now referred to as the Skorohod metric J1. It is defined as follows:
J1(x, y) := inf{ sup P(x(t ), y(A(t))) + sup `t - A(t)I: A E A),
astsb
'
a5t- 1 and An - A. Since An is nonfull, for each n > 1, there is yn E Rd such that II yn II = 1 and r1Ynµn is degenerate on R'. Taking a subsequence if necessary, we may assume that yn -. (some) y. Then 11y,µn - IIyµ (cf. Corollary 1.7.3), and 171Yµ is degenerate, that is, µ is nonfull. Therefore, F is an open set. Suppose there are p., v E .F(Rd) with µ * v 14 'F(Rd). Proposition
2.1.1 gives that there is y # 0 such that J 2(ty)I I v(ty)I = 1 for all t E R. Hence, we have Iii(ty)I = Ii(ty)I = 1, that is, µ and v are both nonfull. Thus, µ, v E F implies µ * v E -49-. Therefore, F is a subsemigroup of 9'(Rd). Q.E.D. Corollary 2.1.3. Let A E End(lBd) and µ E 9(68d). Then All is full if and only if A is invertible and µ is full.
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
48
Proof. For the sufficiency, suppose Aµ is nonfull. Then for some
y' 0, Hy(Aµ) = 3(a) for some a E E. Since A E Aut, A*y
0.
But,
S(a) = H (Ap) = HA-y(µ)' Therefore, µ is nonfull.
For the necessity, suppose either A 0 Aut or µe,9. When A 0 Aut, there is y # 0 such that A*y = 0. Consequently,
3(0) = nA*y(µ) = ny(Aµ), so Aµ is nonfull. When µ is nonfull, we have supp(Aµ) = A(supp µ) Q.E.D. is contained in a proper hyperspace of 084, so Aµ is nonfull. The preceding corollary may be stated in terms of affine transformations.
Corollary 2.1.4. Let a be an affine transformation on 1 d and let µ E 9(1184). Then aµ is full if and only if a is invertible and µ is full.
2.2 CONVERGENCE OF TYPES THEOREMS
We denote the affine transformation a: x -> Ax + a by (A; a), where
A E End(Rd) and a E Rd. By V(Rd) and(Rd) we denote the sets of all affine and of all invertible affine transformations on Rd, respec-
tively; briefly, we sometimes write sV or V,. The transformation (A; a) is invertible if and only if A is invertible. With composition as the operation, sad becomes a semigroup and sad a group of transformations. Defining for a = (A; a) E sad, /3 = (B; b) e sad, and r E 08,
a + J3 = (A + B; a + b), and ra = (rA; ra), we obtain that sad is a real linear space. Also, . becomes a Banach space using the norm II(A; a)ll = max(IIAII, Hall), where IIAII and Hall are some norm on End(08d) and some norm on R4; all norms on finite-dimensional linear spaces are equivalent. This norm gives sV a topology under which composition is continuous.
If µ µ in 9 and a,, - a in sat, then aµ
in 9 (cf.
Corollary 1.7.4). Furthermore, since F(08 d) is open in 69(04) (cf. Corollary 2.1.2), we obtain from Corollary 2.1.4 the following corollary.
49
CONVERGENCE OF TYPES THEOREMS
If anµn µ with µn E .9, µ E .IF, and an E V, then an E V, and An is full for all sufficiently large n. Corollary 2.2.1.
In this section, we investigate the following problem. Let µn - µ and anµn - v. What can one say about (an), µ, and v? We begin with the following auxiliary lemmas. Lemma 2.2.2. Let µn, µ E ,9(IlBd), yn E 1 d, and rn e R. If An µ with µ full, and if {II3,µn * S(rn)} is tight on R, then
and
Supllynll < oo
suplrnl < 00.
Proof. Let sn = llynll + Irnl. Suppose {sn) is unbounded. Choose a subsequence {sn) tending to infinity. Since the balls are compact in a finite-dimensional linear space, we may choose a further subsequence such that Y,
--> yo
rn
and
Sn
Sn
-* ro with Ilyoll + Irol = 1,
(2.2.1)
suppressing the primes on n for notational convenience. This together with tightness gives
7 11 Y./S. µn * V
rn 77'-7
I 11S,.µ * S(ro).
(2.2.2)
Sn
On the other hand, since (I1y,µn * 8(rn)) is tight and sn - oo, we infer that 11y"/S"µn
*S(r"" =T 1/S^(IIvnAn *S(rn)) -S(0)
(2.2.3))
111
(cf. Corollary 1.7.3), where TQ is scalar multiplication by a. But, (2.2.2)
and (2.2.3) together imply that H ,oµ = S(-ro). From (2.2.1) we know
that yo and ro cannot both be equal to zero. Thus, yo # 0 and, by Proposition 2.1.1, µ is nonfull. This contradiction shows that (sn) is bounded.
Lemma 2.23.
Q.E.D.
If µn
µ with µ full and if {anµn} is tight, where
an E QV, then supllanll < oo, that is, (an) is conditionally compact in Qf.
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
50
Proof. Let an = (An; an). Then, (for x, y ER , HIy(anx) = flA*y(x) + fly(an), SO
riy(anAn) = IIAnyµn * S(rlyan)'
Since (anµn) is tight, for each s > 0, there is a compact set K. such that for all n, anµn(KE) > 1 - e. Since fly is continuous, IIy(KE) is compact. Note that fly'(fly(KE)) D K. Thus, (Hyanµn)(1IyKE) >- anp. (KE) > 1 - e for all n. Therefore, (IIy(anµn)) is tight. By Lemma 2.2.2, suplIA*,yll < oo and supllllyanll < oD for all y E R'. Hence, suplIAnll < oo and supllanll < 00, that is, supllanll < oo. Q.E.D.
Corollary 2.2.4. tionally compact in
Let µ be full and an E Q f. (anµ) is condiif and only if {an) is conditionally compact in '.
In the convergence of types theorems, a fundamental role is played by the set of operators having the property that the limit measure µ is unchanged by the action of one of these operators. More formally, we define the invariant semigroup of µ, Inv(µ), to be
Inv(µ) = (a E.1:µ =aµ). Note that the identity transformation is always in Inv(µ), so Inv(µ) is never empty. Also, Inv(µ) is clearly closed under composition. The
next result gives a more precise description of the properties of Inv(µ). Theorem 2.2.5. If µ is full, then Inv(µ) is a compact subgroup of .V,. Conversely, if µ is nonfull, then Inv(µ) is neither a group nor compact.
Proof. Assume a E Inv(µ) and a is not invertible. Since a = (A; a) is not invertible, neither is A. Thus, there is y # 0 such that A*y = 0. Hence, Ilyµ = HHA.yµ * 5((a, y)) = S((a, y)),
so by Proposition 2.1.1, µ is nonfull. Therefore, µ is full implies
51
CONVERGENCE OF TYPES THEOREMS
Inv(µ) c 2f,. Clearly, a' E Inv(µ) for each a E Inv(µ). This estab-
lishes that Inv(A) is a subgroup of .,. To show that Inv(p) is compact, it suffices to show that Inv(µ) is closed and bounded, since we are in a finite-dimensional space. The closedness follows from the
fact that an a Inv(A) with an -) a implies µ = anµ - aµ, so a E Inv(µ). To show Inv(A) is bounded, let (an) be an arbitrary sequence
in Inv(µ). Then anµ = µ for all n, so by Lemma 2.2.3, we have supIlanll < oo. Thus, Inv(p) is bounded. This establishes the first part of the theorem.
Now, assume µ is nonfull. Without loss of generality, we may assume, and do, that µ is concentrated on a proper subspace W of R. Let P be the orthogonal projection onto W For each c c- IR set
Ac = I - P + cP. Then AcW = W, so Acp, = µ, that is, if ac = (Ar; 0), then ac E Inv(µ). Since Ila,ll = III - PIT + Icl IIPII > Icl, we see that Inv(p.) is unbounded. Hence, Inv(A) cannot be compact. Also, since I - P e Inv(p) and is not invertible, Inv(µ) is not a group. Q.E.D.
Lemma 2.2.6.
Let µ E .9 and a E .Qf,. Then
Inv(a,u) = a(Inv(µ))a-'. Proof. The following sequence of statements are equivalent and establish the lemma. /3 E Inv(aµ) if and only if aµ = (3(aµ) if and
only if µ = (a -'/3a)µ if and only if a-'/ia E Inv(µ) if and only if /3 E a(Inv(µ))a-'. Q.E.D. We say that two measures µ and v are of the same operator type provided there is a E Q/ such that µ = av. For full measures, "being of the same operator type" is an equivalence relation. Of importance is the relationship between two measures µ and v when F'nµ'n µ and anµn - v; note the same sequence (µn). The convergence of types theorem states that for full measures µ and v, it is necessary and sufficient that they be of the same operator type. Furthermore, it gives a relationship between the norming sequence (/3n) and (an) which involves the invariant group of A. Preliminary to the convergence of types theorem, we deal with the following special case. Theorem 2.2.7. Assume that µn - p. with µ full. Then in order that anµn - µ, it is necessary and sufficient that an have the form
an = 71nyn for sufficiently large n, where r7n - 770 = (I; 0) and yn E Inv(µ).
52
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
Proof For the sufficiency, by Theorem 2.2.5, (yn) is precompact in 770, {an} is also .ice and all of its limit points are in Inv(µ). Since 77n
precompact in sad and all of its limit points are in Inv(µ). Since µn - µ and any subsequence of (an) contains a further subsequence converging to some a E Inv(µ), we see that anµn µ. For the necessity, by Lemma 2.2.3, we have that (an) is conditionally compact in V. Let F denote the set of all limit points of the
sequence (an). We now show that F c Inv(µ). Let a E F. Then an, - a for some subsequence {an,) of (an). Thus, a,; µn. - aµ and an,µ µ, so µ = all, that is, a E Inv(µ). Since µ is full, each a e F is invertible (cf. Corollary 2.1.4). Since F is a closed subset of a
compact set, F is compact. Therefore, for each n > 1, there is a yn E F such that min{IIan - yll: y E F) = Ilan - ynll. Note that IIan 0. Setting iin = anyrt 1, we have an = ?7n-/n and
05 II17n-770II I as n - , for each n, B,k -> 0 as k -
and the set {Bn k:
k >- 0, n >: 11 is conditionally compact in End(f{8d), then ji never vanishes.
Proof. Let An E .9 be such that µ = Bnµ * µn. Since 2(y) = W(B*Y)An(Y)
and Bn - I, we have µn(y) - 1 in some neighborhood of the origin. Hence, An - 3(0) since {µn} is conditionally compact. Therefore, µn(y) -* 1 uniformly on compact sets.
Now, suppose to the contrary of the proposition that there is a E Rd such that µ,(a) = 0 and µ(y) 0 0 for all y with IIYIl < Ilall. Let C
closure{B*ka: k >: 0, n >: 11.
58
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
Since C is compact,
lira sup l1 -
n-w yeC
I = 0.
Hence, we may assume that µn(y) 0 0 for all y e C and for all n > 1. Thus, if y e C and µ(y) = 0, then µ(B,* y) = 0, because µ(y) = µ(B*y)µn(y). Since a E C and µ(a) = 0, µ(B*a) = 0 for all n > 1. Since B*a E C, µ(B,*2a) = 0 for all n >- 1. Continuing this argument, we see that µ vanishes on (B*ka: k >t 0, n >- 1). By continuity, µ vanishes on all of C. But, 0 E C. This contradiction establishes the proposition.
Q.E.D.
The Gaussian measures are always of particular interest. We characterize the Urbanik semigroups and give sufficient conditions in terms of D(v) for v to be Gaussian. Before stating them, we recall that v is Gaussian, if and only if, whenever v = v, * v2, both v, and v2 are Gaussian, by the Cramer theorem (cf. Theorem 1.8.18). More-
over, the covariance operator of v is the sum of the covariance operators of v, and v2. Furthermore, if A E End(l8d) and if R is the covariance operator of v, then Av is again Gaussian with covariance operator ARA*. Proposition 2.3.8.
Let v be Gaussian with covariance operator R.
Then
D(v) = (A E End(l '): R - ARA* is positive-semidefinite). Proof. Let A E D(v). Then there is vA E 9 such that v = Av * PA. Hence, vA is Gaussian by the Cramer theorem, and its covariance operator, R - ARA*, is positive-semidefinite. Conversely, if A E End(Rd) is such that R - ARA* is positivesemidefinite, then there is a Gaussian measure vA with covariance
operator R - ARA* (cf. Theorem 1.8.6). Note that R - ARA* is symmetric. Consequently, A E D(v).
Q.E.D.
Proposition 23.9. Let v (-= .9Rd). Assume D(v) contains orthogonal projections J,, ... , Jd, Q1, Q2 satisfying the conditions J;Jj = 0 for all i * j, Q1 Q2 = 0, and QkJ, * 0 for all k and i. Then v is Gaussian.
Proof. This is a direct application of the Skitovich theorem (cf. Theorem 1.8.19).
Q.E.D.
COMPACT SUBGROUPS OF .'(Rd) AND Aut(Rd)
59
Proposition 23.10. Let A be a full Gaussian measure on Rd. Then, for some W E Aut(Rd),
A(A) = W-'®W, where d is the group of all orthogonal transformations.
Proof. Let W E Aut(l ) be such that, for some w E Rd, Wµ * S(w) has the property that any orthogonal projection of Wµ * S(w) onto any one-dimensional subspace of Rd is a standard Gaussian measure. Then A(WA) = 6. Hence, by Lemma 2.3.3, WA(A)W-' = B, that is, Q.E.D. A(A) = W-'6W. We conclude this section with some clcmentary properties of D(A).
Proposition 23.11 (a) D(A) = D(A * S(x)) for all vectors x. (b) D(A1) fl D(µ2) S D(µ1 * µ2). (c) If B E A(A), then D(BA) = D(A).
(d) For a > 0 and A E ID, A(A") = A(A) and D(µ") = D(A). Proof
(a) Since A = AA * v if and only if A * S(x) = A(A * 5(x)) * v * S(x - Ax) for all vectors x, we see that D(A) = D(A * S(x)). (b) If Al = AA1 * v1 and A2 = AA2 * v2, then 91 * A2 = A(µ1 * µ2) * v1 * v21 so D(A1) fl D(µ2) c D(µ1 * µ2).
(c) If B E A(A), then BA = A * S(b) for some b. Hence, D(BA) _
D(A * S(b)) = D(A), by (a). (d) If B E A(A), then A" = BA" * S(ab) for some b, so B E A(A").
The reverse inclusion is done in the same manner. Thus, A(A") = A(A). Similarly, D(W) = D(A).
Q.E.D.
2.4 COMPACT SUBGROUPS OF.r,(Rd) AND Aut(Rd) With a probability measure A on Rd we associated the following sets of mappings: the invariant semigroup Inv(A) of affine transformations; the Urbanik semigroup D(A) of linear operators; and the symmetry semigroup A(A) of linear operators. In case A is full, we know that
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
60
Inv(A) is a compact subgroup of sV,(R ), D(µ) is a compact subsemigroup of End(U8d), and A(µ) is a compact subgroup of Aut(R'). This suggests the problem of how to characterize those compact subgroups of sat', or compact subsemigroup of End which are the invariant groups or decomposability semigroups of some probability measures. We are not going to investigate this important problem but only recall a classical theorem on compact subgroups of sV,(R") which we need later on. First, we note that not every compact subgroup of sat', can be the
invariant group for some A. For instance, the compact group of rotations on W' cannot be the invariant group for any measure A. To see this, assume µ has the group of rotations contained in its invariant group. For any y E fl and any one-dimensional subspace W, there is a rotation taking y into W. Hence, µ is a function only of the norm of
y. Thus, its invariant group also contains all reflections about any subspace.
Let ® denote the group of real orthogonal operators on Ind. For A E ®, we have II All = 1 and A ' = A*, so ® is a compact subgroup of Aut(O ').
Theorem 2.4.1.
Any compact subgroup .1 of sad, has the following
form: there exist a closed (hence compact) subgroup O, of 6, a point x0 E Rd, and a symmetric positive-definite operator V such that .' consists of exactly those affine transformations a such that
ax = VWV-'(x - x0) + x0,
We ®p .
Proof. For a = (A; a), set 1(a) = det A. From the formulas
a 'x = A"x +A n-'a + -
+Aa + a for n >_ 1
and a-'x = A-'x - A-'a, we obtain I(an) = (I(a))" and I(a-') = (1(a))-'. Since I( ) is continuous in the topology of sW, it is bounded on the compact group vim. It follows that I(a) is either I or -1 for any a E-= _V.
Let L be any bounded open set in l
and set
M := U a(L). Ot E.
61
COMPACT SUBGROUPS OF *(Rd) AND Aut(68d)
Then M is bounded and open, and aM = M for all a in J. Let xo := IM1 ' fmxdx E Rd,
where IMI denotes the Lebesgue measure of M. Since for a E .# the Jacobian is II(a)I = 1 and since aM = M, we have axo = xo for all a E ,4. Therefore, Y consists of all affine transformations a of the form
a(x) = A(x - x0) + x0 for A E .ei, where -e, is a compact subgroup of Aut(l ). Let L be bounded and open, as before. Then the set
N:= AUe de,A*(L) is bounded and open, and A*N = N for all A E _e,. Let U be the bounded linear operator defined by (x, Uy)
fN (x, z)(y, z) dz.
Hence, U is symmetric and (x, Ux) >- 0. Moreover, (x, Ux) = 0 implies (x, z) = 0 for all z in N, and since N is open, x = 0. Thus, U is a symmetric positive-definite operator, so U has a square root V, that is, V2 = U, which is also a symmetric positive-definite operator.
Since for A E f, the determinant of A is either plus or minus one and since N is A*-invariant, we have, for all A E ,, (Ax, UAy) = (VAx, VAy) = fN (x, u)(y, u) d(A*-'u) = (Vx Vy)
Setting x = V-'u and y = V-'v for u, v E Rd, we see that VAV-' is orthogonal for each A E .0,. Finally, the family (VAV-': A E S,) forms a compact subgroup ®0 of the orthogonal group. Corollary 2.4.2.
Q.E.D.
Any compact subgroup f of Aut(68d) has the form
.1= Vd0V-1, where 6o is a closed subgroup of the group of all orthogonal operators on Rd and V is a symmetric positive-definite operator.
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
62
2.5 EXAMPLES IN INFINITE-DIMENSIONAL SPACES
In this section, H denotes an infinite-dimensional real separable Hilbert space with complete orthonormal basis (en}. Thus, each vector
x E H has a unique representation x = Exne,,, where xn E W and 11x112 = Ext. More precisely, we have xn = (x, en) for n z 1. Example 2.5.1 shows that the convergence of the characteristic functions does not imply the convergence of the corresponding measures. This is Example 1.7.7, again. Example 2.5.2 shows that Lemma 2.2.3 and Theorem 2.2.7 do not hold in H. Example 2.5.3 shows that the set of all full measures .51-(H) is not open in 9(H), that is, Corollary 2.1.2 does not extend to H. Example 2.5.4 shows that Inv(µ) need neither
be compact nor a group, even when µ is a full Gaussian measure. Example 2.5.5 shows that the convergence of types theorem 2.2.10 does not hold in H.
S(en) and µ
S(0). Then for each y r= H, µn(y) = exp i (y, en) --+ 1 = µ(y) as n - oo. It is easy to see that for point measures S(xn) to converge to S(x) it is necessary and sufficient that xn -' X. Here, Hen - e .. 11 =' for all n * m. Hence, Example 2.5.1.
Let µn
S(en) cannot converge to 8(0). Therefore, the continuity theorem does not hold in infinite-dimensional spaces. Example 2.5.2. Let an > 0 with Ean = 1, and let
Ean5(en).
Then µ is full on H. Let An: H -> H be given by Anx = F, xiei + nxnen, ion
when x = Exiei. Then each Anis a bounded linear invertible operator with IIAnII = n. For arbitrary f: H -, R' bounded and continuous, we have
f f(x)(Anµ)(dx) = Li aif(ei) + anf(nen) -* Eaif(ei) ion
= f f(x)A(dx),
so Anµ µ. Since IIAnII = n, we see that Theorem 2.2.7 and Lemma 2.2.3 do not hold in infinite-dimensional Hilbert spaces.
EXAMPLES IN INFINITE-DIMENSIONAL SPACES
Example 2.5.3.
63
Let n
x; e;
Pn x
for x = E x, e; .
i=1
µ for all p. E .9(H).
Clearly, IIPn - III -- 0 as n -> co, so Pnµ
Since the Pnµ are concentrated in finite-dimensional subspaces of H, we see that .IF(H) is not open in .9(H) (cf. Corollary 2.1.2). Example 2.5.4.
Define D on H by Dx
Ek-2Xkek.
Then D is a positive-definite Hermitian operator with finite trace. Let
Anx = F, (n + k)k-Ixn+kek for n Z 1. Hence,
k(k -
A*,x =
n)-lxk-nek
k=n+1
L.. xkek,
k=1
where Xk is 0 for 1 < k < n and is k(k - n)-lxk_n for k > n.
Consequently, for x, y E H, (AnDAnx, y) _ (DAnx, Any) W
=
F, (k - n)
z
Xk_nyk_n = (Dx, y),
k=n+1
so AnDA*, = D. Let lk be the symmetric Gaussian measure with the operator D for its covariance operator (cf. Theorem 1.8.6). Then µ is
full (cf. Theorem 1.8.20), and (An; 0) E Inv(µ). But, each An is noninvertible and IIAnJI = n + 1 --> oo. Therefore, Inv(µ) is neither compact nor a group although µ is full (cf. Theorem 2.2.5). Example 2.5.5.
Let v
FanS(en),
64
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
where an > 0 and Ean = 1. Also, let e
in := " and N- := ranb(en). n Let n
m
Anx = E kXkek +
xkek.
k-n+l
k=1
Then IIAnII = n, An are invertible, and for bounded continuous f we have 00
f f(x)(Anp)(dx) = k akf(k-Anek) k=1 n
F, akf(ek) +
k=1
=
W
T, akf(k-'ek)
k=n+1
-F, akf(ek)
f f(x)v(dx)
Therefore, A, IL - v, µ and v are full measures, and IIAnII = n (cf. Lemma 2.2.3). Now, we show that there does not exist a sequence (Bn) of bounded linear operators on H such that Bnµ v and supllBnll < oo. Thus, in particular, there is no bounded linear operator B such that
Bµ = v. To the contrary, assume Bnµ = v and supllBnjj < r, with r an integer. To obtain a contradiction, we find a bounded continuous f
such that ff(X)(Bnµ)(dx) does not converge to ff(x)v(dx). Define
f(x)
min{1, Ilxll}. Then
ff(x)v(dx) = Fanf(en) =
1.
On the other hand, for k > 2r, we have f (Bnek) = IIBnekII < 1/2 for all n >_ 1. Hence,
ff(x)(Bnµ)(dx)
kFlak +
2 k=2r+lak < 1,
that is, ff(x)(Bnµ)(dx) is bounded away from one.
THE CASE WHEN d = 1; COMMENTS
65
This concludes the examples. All the above examples show that in the case of operator-limit theorems in infinite-dimensional spaces we must add some special assumptions on the norming sequences.
2.6 THE CASE WHEN d = 1; COMMENTS
In this section, we consider the case opposite to the one in the previous section. Namely, we assume that d = 1 and we work with probabilities on the real line. The idea of fullness reduces to nondegeneracy, that is, µ is degenerate if and only if µ = 6(a) for some
aEW.
If a = (a; b) is an affine transformation on R', that is, ax = ax + b, and if a E Inv(µ), then
I#(t)I =Iµ(tan)I for all t E l' and for all n >- 1. Hence, when µ is full, Jal = 1. Furthermore, letting Tax = ax, we have µ = T_ lµ * 6(b) for some b E R' if and only if µ is a translate by 6(b/2) of a symmetric measure v, that is, µ = v * 6(b/2). Hence, either
Inv(µ) = {(1;0)} or Inv(µ) = {(1;0),( - 1;b)}; note that ( - 1; b)2 = (1; 0). Consequently, we have that the symme-
try group A(µ) of a nondegenerate µ is either (1) when µ is not a translate of a symmetric measure or (1, - 1) when µ is. From this, we now obtain the classical convergence of types theorem on R'. Corollary 2.6.1. If µn - µ and Tanµn * 5(bn) - µ, where µ is nondegenerate, an > 0, and bn E R', then an -* 1 and bn -' 0.
Proof. Let an = (an; bn). By Theorem 2.2.7, we have an = -gnyn
for some ?]n - (1; 0) and yn E Inv(µ). Since an > 0, we see that yn = (1; 0) for sufficiently large n. Consequently, an -* 1 and bn -* 0. Q.E.D.
Corollary 2.6.2.
If µn - µ and Ta.µn * S(bn) - v,
where µ and v are nondegenerate, an > 0 and bn c- R1, then there are
66
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
a>0and be68'such that an ->a,bn->b,and v = Taµ * s(b).
Proof. From Theorem 2.2.7 and Lemma 2.2.9, we have that an .= (an; bn) is conditionally compact in d, and there is a = (a; b) with a > 0 such that v = aµ. Consequently, Theorem 2.2.10 implies an = where yn E Inv(µ) and 71n -> (1; 0). Thus, yn = (1; 0) and Q.E.D. an -* a.
Let us now consider the Urbanik semigroups on R. If for some a e 118' and some P. E 60W), we have l = T. IL - 1L.,
then N
=Tanµ*va,n
where van = TaA. * T(az)µa * ... * T,(a")E'r'a
Hence, =1µ(1)I
so if a > 1, we see that
1 =Iµ(t)IIi(t)I for all t, where v is the limit of van as n - oo, that is, µ is degenerate. Similarly, if a < - 1, taking a subsequence we again see that µ must be degenerate. Therefore, for µ nondegenerate, we have
D(µ) c [-1,1]. Also, D(µ) is a compact semigroup by Theorem 2.3.1. From Proposition 2.3.8, we see that when v is Gaussian, we have
D(v) = [ -1,1] .
67
STANDARD CONVERGENCE OF TYPES
Further examples of the decomposability semigroups on I8' are presented in Urbanik (1976). Ilinskii (1978) proved that each compact subsemigroup D of [0, 1] with 0, 1 E D is a decomposability semigroup
of some measure on R. Some further results in this area are also given in Niedbalsha-Rajba (1981).
2.7 STANDARD CONVERGENCE OF TYPES
The examples discussed in Section 2.5 show how much operator type
convergence theorems depend on the dimension of the underlying space and on fullness of the measures in question. Roughly speaking, when we allow normalization to be by arbitrary affine transformations,
then we have to consider only full measures on finite-dimensional linear spaces. In this section, we will impose restrictions on the form of affine mappings (arbitrary lines); on the other hand, we relax conditions on the measures (nondegenerate, i.e., not concentrated at a single vector) and the underlying space (arbitrary Banach space). Let X be a Banach space. Affine mappings a of the form
a(x):=ax+v,
a e R',v,xEX,
(2.7.1)
are called lines in X. Lines with a 0 0 form a subgroup in Aff(X) with composition as the operator. Mappings a given by (2.7.1) will be written as
a(x) = Tax + v.
(2.7.2)
Measures µ, v r= 90(X) are said to be of the same type if there exist a c- Q8' and v E X such that Tav * 5(v),
(2.7.3)
where 8(v) is the degenerate measure concentrated at v e X. Let us note that degenerate measures form a closed (in weak convergence topology) subsemigroup of -60(X) (cf. Remark 1.6.7). Also, one usually assumes that a > 0 in (2.7.3) in order to obtain the facts below. At the end of this section, we consider arbitrary a E R. Proposition 2.7.1. Let X be a Banach space and assume µ - µ in .9'(X ), where µ is nondegenerate. If a,, > 0, v,, E X, then
T.,µ, *
if and only if a -* 1 and v -> 0 in X.
68
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
Proof. The direct part is a particular case of Corollary 1.7.4. For the converse part, taking symmetrizations we have µ° 0 3(0), µn - µ°, Tµ° - µ° as n -> oo. and pn If zero is a limit point of (an), then Corollary 1.7.4 implies µ° = 8(0). Since TI/a" pn - µ°, infinity cannot be a limit point of (an). Finally, if
c, and c2 are two limit points of {an), then Corollary 1.7.4 gives µ°=TcIMP=T /h° and hence
n = 1,2,... . Since µ° # 6(0), we have c,/c2 = 1. Consequently, an -> 1. To show that vn - 0, first observe that (vn) is conditionally compact in X, by Theorem 1.7.1(b). Furthermore, if w is a limit point of (vn), then we have µ * 3(w) = µ. Taking Fourier transforms, we arrive at exp i (x *, (o) = 1 in some open neighborhood U of zero in X*. Thus, for x* E U, there exists
an integer k such that (x*,(o) = 2irk(x*). Since k(0) = 0 and is continuous, we conclude that k(x*) = 0 for x* E U. Consequently,
(x*, (o) = 0 for all x* E X* and therefore w = 0. This proves that v --> 0 in X.
Q.E.D.
Remark. The above proposition is an analog of Theorem 2.2.7 where one deals with operator types and full measures on QBd. Also, see Example 2.5.1 for infinite-dimensional linear spaces. Corollary 2.7.2. Then T,,µ,, * S(un)
Let µn = p. in .9'(X), with µ nondegenerate.
v for some c > 0, un E X, with nondegenerate v if and only if cn - co > 0, un -* u° in X, and v = Tcoµ * 3(uo). Proof. First, taking symmetrizations and reasoning as in the above proof of Proposition 2.7.1, we conclude that cn --> co > 0. This implies
that (u,,) is conditionally compact in X and once again repeating the
arguments from the above, we obtain un -> uo in X. Finally, we appeal to Corollary 1.7.4.
Q.E.D.
Corollary 2.7.3 (Convergence of Types). Assume Ta An * S(vn) µ, µ is nondegenerate, and an > 0, vn E X. In order for Tbµn * S(wn) V, with v nondegenerate, it is necessary and sufficient that bnan 3 -' co > 0, wn - bnan 'vn -> uo in X, and v = Tcoµ * 6(uo).
69
STANDARD CONVERGENCE OF TYPES
Proof. Note that
S(un) = T,
a;i(T..A'n *
5(vn) * 3(wn - bn an 'v,,) Q.E.D.
and apply Corollary 2.7.2.
Now, let us consider normalization Ta with a E R, not necessarily positive.
Lemma 2.7.4. If An - µ and Tanµn * S(un) - µ with µ nondegenerate and an E IfRI, un E X, then lanl - 1, un, -> 0 when n' E (n: an > 0}, and u,;. -> u° when n" E (n: an < 0). Moreover, if -1 is a limit point of {an}, then µ is the translation of a symmetric measure.
Proof. Taking symmetrizations we have
µ - µ°
and
T,, 1
- µ° 0 S(0).
(2.7.4)
As in the proof of Proposition 2.7.1, we see that neither 0 nor ±oo can (an}. Furthermore, if c is a limit point of (an) and be limit points of 0 < Icl < oo, then (2.7.4) and Corollary 1.7.4 give TTµ° = µ°
and
µ° = Tµ° = Ticinµ°,
n > 1.
Thus, Icl = 1 and land --> 1. Hence, {Ta An} is conditionally compact in .60(X) and, by Theorem 1.7.1, so is {u;,} in X. Let (n') and {n"} be subsequences corresponding to the positive and the negative terms of (an), respectively. Note an # 0 for all sufficiently
large n. If {n'} is infinite, then, by Proposition 2.7.1, u -p 0. In the opposite case, when (n") is infinite, suppose wl and w2 are limit points of (un.}, then we obtain
T-lµ * 5(w1) = A = T-1N- * 5(w2) Hence, w, = w2, that is, un. --> u° for some u° E X. From the above, Q.E.D. we also conclude the last part of the lemma. Remark 2.7.5. If -1 is a limit point of (an), then v = µ * 5(x) is symmetric for some x E X. Thus, our setting can be changed to the following one. If An v and T,, An * S(wn) - v, with v nondegenerate and sym-
metric, then we have lanl - 1 and w" -* 0. Also, the converse is true.
70
CONVERGENCE OF TYPES THEOREMS, SYMMETRY GROUPS
So, we are in the same position as in Proposition 2.7.1 when symmetric nondegenerate measures are limits.
2.8 BIBLIOGRAPHIC COMMENTS Proposition 2.1.1 is from Sharpe (1969). The material on the convergence of operator types in Section 2.2 is based on Billingsley (1966).
Some of these results were also proved in Sharpe (1969) and in Weissman (1976). In fact, the Khintchine theorem was proved by Fisz as early as 1954. All the results in Section 2.3 are due to Urbanik. We collect them from his papers: (1972), (1975), (1976), (1978), and (1979). Characterization of the compact affine groups on Rd, Section 2.4, is well known. Originally, it is due to Fenchal (1936), but we quote its proof after Billingsley (1966). Examples in Section 2.5 are from Jurek (1981). They show the importance of fullness of the measures and of the finite dimensionality of the linear spaces. Results in Section 2.7 are well known and standard, except Lemma 2.7.4 which seems not to be stated explicitly elsewhere.
CHAPTER 3
Operator-Selfdecomposable Measures
The Urbanik decomposability semigroup D(µ) of a measure µ is the set of all linear operators A such that p. = Aµ * v for some measure P. In Section 3.3, it is shown that for a full operator-selfdecomposable measure it, D(µ) always contains a one-parameter semigroup of the form {exp(-tQ): t >- 0} for some linear operator Q such that exp(-tQ) - 0 as t -a 00. In fact, this condition is sufficient for µ to be operator-selfdecomposable. Preliminary to this, some basic properties of norming sequences are obtained in Section 3.2. Section 3.4 deals with the representation of the characteristic function of a measure whose decomposability semigroup contains the one-parameter semigroup {exp(-tQ): t >- 01. A useful new norm associated with this semigroup is introduced. Section 3.5 is concerned with the question of the generators for the class L0(Q) of all µ whose semigroup D(µ)
contains (exp(-tQ): t >t 0) for a fixed Q. The result obtained is analogous to the characterization of the class of all infinitely divisible
laws as the smallest class closed with respect to weak limits which contains the compound Poisson laws. In Section 3.6, a random integral representation is obtained for these exp(-tQ)-decomposable laws, that is, laws in L0(Q). This random integral representation shows that any measure in LO(Q) is the limit in distribution of a particular Markov process as time goes to infinity. Section 3.7 deals with the corresponding Markov semigroups. A characterization of their infinitesimal generators is obtained. In Section 3.8, it is shown that all full measures in L0(Q) are absolutely continuous. 3.1
STATEMENT OF THE PROBLEM
Let (e,) be a sequence of independent ll -valued random variables and let {An} be a sequence of bounded linear operators on R". 71
OPERATOR-SELFDECOMPOSABLE MEASURES
72
Assume that the triangular array {An6k: 1 _< k _< n, 1 _ 1) is infinitesimal and * An)* S(an) - A. For p. full, we may assume that every An(E'1'1 * norming sequence is in Aut. ..
73
NORMING SEQUENCES
Proposition 3.2.1. For every norming sequence {An} of a full operator-selfdecomposable measure µ, we have
lim An = 0. n--
Proof. Suppose to the contrary that for some sequence {An} the proposition fails to hold. Without loss of generality, we assume Ajg1 * ... * µn) * S(an) - µ and lim IIAnII > 0.
n-w
Select vectors z,, so that Ilznll = 1 and IIA*II = IIA*,znll for all n Set un '= IIA*II-'A*zn
for n >_ 1.
Then Ilunll = 1, so without loss of generality we assume un --> (some) u
with (lull = 1.
By the infinitesimal assumption, Anµj - 8(0) for each j >_ 1. This is bounded
together with the fact that the sequence implies that, for all c E p8' and all j >_ 1,
ryim (Anµj)-(cllA*,II`'zn) = 1.
On the other hand, we have (Anp'j)A
(CllAnll-'z,,) = j2 (cun) -> 11j(cu).
Thus, for all c e R' and all j ? 1, µj(cu) = 1
with u * 0.
Hence, n
* µn)" (cu) = fµj(cu) = 1 l=1
for all c E l
and all n >_ 1, with u : 0. This implies that µ is nonfull
OPERATOR-SELFDECOMPOSABLE MEASURES
74
(cf. Corollaries 2.1.2 and 2.1.3). This contradiction establishes the Q.E.D.
proposition.
Proposition 3.2.2. For every full operator-selfdecomposable measure there is a norming sequence (An) such that
lim An+lAn' = I.
n-oo
Proof. Let pn = Bnvn * S(an) with vn '= E'1'1 * ... *An and assume pn - µ and {Bnµk: 1 < k 5 n, n >_ 1) is infinitesimal. Since pn+1 = Bn+1Bn(Bnvn * S(an)) * Bn+1p'n+1 * S(Cn)
with Cn.- an+1 - Bn+1Bn
and since Bn1An,
1
1an
S(0), we have
Bn+1Bn'pn * 6(Cn) - µ.
(3.2.1)
Consequently, by Theorem 2.2.6, (Bn+1Bn') is conditionally compact in Aut and the set T of all its limit points is contained in the compact symmetry group A(µ). For each n >_ 1, choose Fn E T such that En = II Fn -
Bn+1Bn'II = min(IIF - Bn+1Bn'II: F E T).
Obviously, En -> 0. Set H1
I and for n >_ 2, set
Hn:=F1 1...Fn 11. Then each Hn E A(µ). Now, set An
HnBn.
Using the compactness of A(µ) and the usual subsequence argument
(cf. Corollary 1.6.3), we see that the sequence Anvn * S(Han) is conditionally compact and all of its limit points are of the form µ * S(b) for some b. Hence, there is a sequence bn such that Anvn * S(bn) - A.
NORMING SEQUENCES
75
Therefore, {An} is a norming sequence for µ. Furthermore, IIAn+IAn' - III _< IIIInII IIHn+1Ilen < en sup2(IIAII: A E A(µ)).
Since en -> 0 and the supremum is finite, we have An+1An' -' I. Q.E.D.
Proposition 3.2.3.
Let (mk) and (nk} be sequences of positive
For every norming sequence (An) integers with Mk _ 1) is infinitesimal. Setting / i Wk := Ank(N'mk+I * ... * Ank) * S ank - Ank`4mkamk),
we have Pnk = AnkAmkpmk * Wk .
(3.2.2)
By Theorem 1.7.1, (AnkAmkpmk) is conditionally compact in End. uO, by Lemma 2.2.3, we have that (AnkAmk) is conditionally compact in End. Let A E End be a limit point of (AnkAmk). Without loss of generality, we assume AnkAmk -> A. Then Since pmk
Pnk - µ and
AnkAmk pmk = *AA.
(3.2.3)
By Theorem 1.7.1, (3.2.2) and (3.2.3) imply that (Wk} is conditionally
compact. Taking a subsequence if necessary, we may assume Wk (some)w. Therefore, by this convergence and (3.2.2) and (3.2.3), we have µ = Aµ * W, that is, A E D(µ). Q.E.D. Earlier, we used sem(F) to denote the smallest closed multiplicative subsemigroup of operators containing F, where F is a subset of End. From Proposition 3.2.3 we get the following corollary. Corollary 3.2.4. If (An) is a norming sequence for a full operatorselfdecomposable measure µ, then 1 < m _< n, n >_ 1} is compact in End.
OPERATOR-SELFDECOMPOSABLE MEASURES
76
Proposition 3.2.5. For every norming sequence (An) of a full operator-selfdecomposable measure µ, we have that
A(µ) n
1 < m < n, n >_ 1)
is a compact group containing all the limit points of the sequence {An+1An 1).
Proof. Let B denote the set of interest. The compactness of B is an immediate consequence of Corollaries 3.2.4 and 2.3.2. From the proof of Proposition 3.2.2 and Theorem 2.2.6, we have that Ai + 1 An 1 =
DnC,,, where Dn -* I and each C E A(µ). Hence, the limit points of {An+1An 1) are contained in B. Next, we show that B is a group. It suffices to show that each B E B
has an inverse in B. Consider a B E B and its compact semigroup sem(B). By the Numakura theorem (cf. Corollary 1.1.3), the set G of all limit points of the sequence (Bn) forms a group and it is the minimal ideal of sem(B). Moreover, sem(B) contains exactly one idempotent J, namely the unit of G. Hence, there is C (=- G c B such that
CB=BC=J. Since J E B c A(µ), µ = Jp. * S(x) for some vector x. But, µ is full.
Therefore, J is invertible. Since J is idempotent, it must be the identity. Therefore, C = B-', so B-' E B.
Q.E.D.
3.3 THE URBANIK SEMIGROUP OF AN OPERATOR-SELFDECOMPOSABLE MEASURE AND ITS EXPONENTS
The main result in this section is that the Urbanik decomposability semigroup of a full operator-selfdecomposable measure always contains a one-parameter semigroup of the form exp(-tQ), t >- 0, with Q invertible and exp(-tQ) -- 0 as t - oc. By Theorem 2.3.1, D(µ) is a compact subsemigroup of End whenever µ is full. The nonzero idempotents in D(µ) play an important role in the theory. The two idempotents, I and 0, are always in D(µ). For a given idempotent
77
URBANIK SEMIGROUP OPERATOR-SELFDECOMPOSABLE MEASURE
J E D(µ), we define Dj(µ) and A j(µ) as follows:
DI(µ) (A E D(µ): AJ = JA = A), A,(µ) :_ (A E D,(µ): Jµ = Aµ * S(x) for some vector x). Clearly, D1(µ) and A,(µ) are closed, hence compact, subsemigroups of D(µ) and DJ(µ), respectively. For A1(µ) we have even more. Lemma 3.3.1. Let µ be full and let J be an idempotent in D(µ). Then A,(µ) is a compact group with the unit J.
Proof. By the definition of D1(µ), J is a unit in A,(µ). Let A E A,(µ). Then sem{A) is a compact semigroup and, from the Numakura
theorem, we see that there are an idempotent JI and an operator B, both in sem{A}, such that
AB=BA=J1. Since JI is in Aj(µ), we have JIJ = JJI = JI and Jµ = J1µ * S(x) for some vector x. Since µ is full, the images of J and JI are the same. Consequently, J = JI and we have AB = BA = J, that is, B is an inverse of A in A1(e ). Thus, Ay(e) is a group with unit J.
Q.E.D.
A result which is analogous to Proposition 2.3.4 is the following lemma.
Let µ be full and let J be an idempotent in D(µ). If A E Dj(µ) and J e sem(A), then A E Aj(µ). Lemma 3.3.2.
Proof. Let k 1 < k2:5 be positive integers such that A'- - J. We may assume that k >_ 2 and that the sequence (A'--') converges to some operator B in sem(A). Then we have
J = AB,
µ = Aµ * µA, and µ = Bµ * µB
for some measures µA and µB. Consequently, Aµ = Jµ *Aµ., so µ = Jµ * AµB * µA. Since JA = A, the last equality implies
Jµ =Jµ*AµB*JµA Hence, I(JµA)^ (y)I = 1 for all y in some open neighborhood of the origin, so JµA = S(x) for some vector x; note that H((JµA)°) contains
OPERATOR-SELFDECOMPOSABLE MEASURES
78
an open set containing the origin (cf. Proposition 2.1.1). Finally, the
equation µ = Aµ * µA gives Jµ = Aµ * S(x), that is, A E A,(µ). Q.E.D.
Now we are ready for the main steps in the construction of the uniformly continuous one-parameter semigroup of the linear operators belonging to D(µ). Since the construction is rather long, we divide it into three steps in the following lemma. Lemma 3.3.3. Let µ be full and operator-selfdecomposable and let J be a nonzero idempotent in D(µ).
(i) For any c in (0, 1), there exists an operator B, in D1(µ) \ Aj(µ) such that c = IIJ - BcII = min(IIJ - GBJI: G E A!(µ))'
(ii) There are a constant a > 0 and a subset (Ed: 0 < d < a) of D1(µ) \ A,(µ) such that d = min(IIJ - GEdII: G E A J(µ)) and each Ed is the limit of a sequence of the form (Bm^), where 0 < c,, -b 0, the positive integers Mn - oo, and Bc comes from part (i) of this lemma.
(iii) The semigroup DJ(µ) contains a one parameter semigroup (J exp(tV): t > 0) with V E End such that JV = VJ = V. Moreover, D,(µ) contains an idempotent J1, possibly 0, J, ' J, J1V = VJ1, and lime .,(J - J,)exp(tV) = 0. Proof
(i) Let
be a fixed norming sequence for u. such that I as n -- cc (cf. Proposition 3.2.2). For integers
mzn - 1, let am,,,
min{IIJ
- GAmAn1JII: G E Aj(A)}.
Since A,(µ) is compact by Lemma 3.3.1, am
Since J E Aj(µ), we have a,,,,, = 0 for all n > 1.
n
are well defined. (3.3.1)
URBANIK SEMIGROUP OPERATOR-SELFDECOMPOSABLE MEASURE
79
The compactness of A,(µ) coupled with the fact that Am --> 0 as m -), oo, by Proposition 3.2.1, gives lim am,n = IIJII >_ 1
m -.oo
for all n z 1.
(3.3.2)
Furthermore, the compactness of A1(µ) and of sem(A,,,An1: 1 5 n < m), by Corollary 3.2.4, implies that, for n 5 m, am+1,n 5 min{IIJ - GAmAn'JII +IIG(Am+1Am1 - I)AmAn'JII: G (=- Aj(µ))
< am n + b3IIAm+1Am' - III,
where b is an upper bound for the norms of the operators in these compact sets. Similarly, for n < m, am,n ,n
am+I,n + b3IIAm+IAmt - III.
Consequently, for positive integers 1 < nm 5 m, we have lim lam+1,n,,, - am,nml = 0.
M-00
(3.3.3)
From (3.3.1) and (3.3.2), we see that given any c in (0, 1), we can find integers 1 5 n 5 mn such that am,,,n < C
0 we have that B,, is not in A,(µ). This establishes part (i) of the lemma. (ii) Let Bc be as in part (i) and for n z 1 set bn
, := min(IIJ - GB' II: G E Aj(µ)}.
Then bI,c = c. Consider the compact semigroup sem{Bc}. From the Numakura theorem, there is an idempotent JJ in sem(Bc). Furthermore,
limsupbn,c >_ min{IIJ - GJJI: G E A,(µ)} n-+oo
and JJ is in D,(µ). Since JJc = JJ and J and JJ commute, J - JJ is also idempotent. Moreover, J * Jc; otherwise, Lemma
3.3.2 implies Bc is in A,(µ), contradicting part (i). Consequently,
IIJ - JJII _1. Set
a := inf{IIJ - GJJII: G (=- A,(µ) and 0 < c < 1}.
Suppose a = 0. Then, by the compactness of A,(µ) and
D,(µ), we could find D in A,(µ) and a limit point R of (JJ: 0 < c < 1) such that J = DR. Since R is idempotent and in D,(µ), we get R = JR = DR = J, that is, J is a limit point of (Jc: 0 < c < 1). But, IIJ - JJII > 1 for all c in (0, 1). Thus, a > 0 and lim sup b,,,, >_ a > 0 for all c in (0,1).
n-z
81
URBANIK SEMIGROUP OPERATOR-SELFDECOMPOSABLE MEASURE
Letb;=max(IIAII: A (=- D(µ)). For n
- 1,
iiJ-cB,"+'11- 1, I bn+],C - bn,cl < b2c.
Hence, for every sequence {mn} of positive integers and every sequence (cn) in (0, 1) with cn - 0, we have lim
bm.,c,,I = 0.
(3.3.5)
Thus, given d in (0, a), we can find positive integers mn and numbers c,, in (0, 1) such that c,, - 0 and bm cn < d
0) is contained in D,(µ).
Now consider the compact semigroup, sem(W). By the Numakura theorem, it contains an idempotent J,. Since V and W commute, J,V = VJ,, and since W = Ed'?, we have J, is in sem(Ed.). Consequently, J # J, because otherwise by Lemma 3.3.2 we would have that Edo is in A,(µ) which is false. Since J - J, = (J - J,)J, we have that the semigroup {(J J,)exp(tV): t >- 0) is conditionally compact. Let 0 < t,, --- oo so that
(J - J,)exp(t,,V) -' H,
(3.3.9)
where H is some element of D,(µ). Without loss of generality, we may assume that
W1'-) =Jexp([t,, V) - H1,
Jexp((t,,
- [tn])V) - H21
where both H, and H2 belong to D,(µ). But, H, is a limit
URBANIK SEMIGROUP OPERATOR-SELFDECOMPOSABLE MEASURE
83
point of (Wn). Hence, J1H1 = HIJI = H1, so (J - JI)HI = 0.
Finally, (3.3.9) implies that (J - JI)HIH2 = H, so H = 0. Therefore, all limit points of {(J - JI)exp(tV): t > 0) as t -30 00 are 0. Q.E.D. Lemma 3.3.4. If JI and J2 are idempotents in D(µ) and J2 E D1(µ), then DJZ(µ) a D1I(µ).
Proof. Let A E D,Z(14). Then AJ2 = J2A = A. Since J2 E D j,(p ), JIJ2 = J2J1 = J2. Hence, A = AJ2 = A(J2J1) = (AJ2)JI = AJI. Similarly, A = J1 A. Therefore, A E D1(µ). Q.E.D.
We are now ready to establish the main characterization of an operator-selfdecomposable measure in terms of its decomposability semigroup. Theorem 3.3.5.
A full measure µ on Rd is operator-decomposable if
and only if its Urbanik decomposability semigroup D(µ) contains a one parameter semigroup (exp(-tQ): t > 0) with Q invertible and
exp(-tQ) - 0 as t - oo. Furthermore, if p. = exp(-tQ)µ * µ, for
t > 0, then each µ., is infinitely divisible.
Proof. Assume µ is full and operator-selfdecomposable. Since I is idempotent and belongs to D(µ), by consecutive applications of Lemma 3.3.3(iii), we obtain a sequence {Jn}o of idempotents with JO = I, a
sequence (V)i in End such that Dj(µ) contains the one-parameter semigroup (Ji exp(tV+1): t >- 0), and
Jivi+1 = li+IJi = V+1, Ji+lli+1 = V+1J1+11 lim (Ji - Ji+1)exp(tVi +1) = 0, Ji+I E Dj,(µ), and
Ji # Ji+
unless
Ji = 0.
From Lemma 3.3.4, D.Jµ) c D1(µ). Hence, J E D1.(µ) for n > i, and so
JnJi = JiJn = Jmax(n,i).
(3.3.10)
Consequently, Ji - Jn are idempotents and either Jn = 0 eventually or 1 for n i. However, {Jn) c D(µ) and D(µ) is compact II Ji -
(cf. Theorem 2.3.1). Thus, J, = 0 eventually. Let r be such that
OPERATOR-SELFDECOMPOSABLE MEASURES
84
J =0 for all n >- r. Set K.
J; _ 1
-J, for 1 < i < r. Since K1 _
J; _ (1 - J;), by Theorem 2.3.6(a), K. E D(µ). From (3.3.10), we have
KkKS = 0 for k # s. Finally, note that K. exp(tV) E D(µ) and K;V = V K1 for 1 < i < r. From Theorem 2.3.6(c), we obtain
F, K; exp(tV) E D(µ). i=1
E;=1K;V"' for m > 1. E;=1K;V, we have Setting Q Consequently, exp(-tQ) = E;s1K; exp(tV) E D(µ) and exp(-tQ) 0 as t --> oo. This last property shows that Jo exp(-tQ) dt is well defined, and a simple calculation shows that Q - = Jo exp(- tQ) dt. Now, for the sufficiency, assume µ is full and D(µ) contains {exp(-tQ): t z 0} for some Q with exp(-tQ) --> 0 as t --> co. Set B := exp(-(1/n)Q). Then for each n >: 1 there is a measure vn such that 1
A=Bnµ*vnFrom Theorem 1.7.1, {vn} is conditionally compact in 9. By Proposition 2.3.7, µ never vanishes. Hence, A(Y) vn(Y)
(3.3.11)
= 12(Bn y)
and Pn(y) -> 1 as n -* oo for all y. Consequently, v - S(0). Set An := exp(-QE;s1(1/j)), n z 1. Also set, .ul
Al 'µ,
An `= An'vn
for n > 2.
(3.3.12)
Since An -i 0, we have Anµj, - S(0) whenever the sequence (jn) is for bounded. When in - with in < n, we have AnAj, = AnA71vln
1) is conditionally compact in Aut and vn
in >: 2. Since
S(0)1.
S(0),
we have AnAj^ Therefore, the triangular array (Anµj: 1 < j < n, n >: 1) is infinitesimal. Finally, from (3.3.11) and (3.3.12), 111(Y) = µ'((`4I 1)* y)
and, for k>:2, ((A P'k(Y) = vk((Ak 1)*Y) =
y)
µ ((A kk-l)*Y))
85
URBANIK SEMIGROUP OPERATOR-SELFDECOMPOSABLE MEASURE
and, consequently, An(u*
* µ'n) = µ, which shows that
Ei
is
operator-selfdecomposable.
Finally, we need to show that if µ = exp(-tQ)µ *A, for t >: 0, with Q such that exp(- tQ) -* 0 as t - oo, then each µ, is infinitely divisible. Given t > 0, select a sequence of positive integers (kn} such that
k
kn>n+1 and
-- t j=n+1 E
1
asn -*oo .
Let An be the operators defined in the proof of the sufficiency. Then
Ak An 1 - exp(-tQ) as n -+ 0. Since
µ = Ak An'p * Ajµn+1 is conditionally compact (cf. Theorem we have {Ak(µn+I * . * 1.7.1). Moreover, since µ never vanishes (cf. Proposition 2.3.7),
(`4k (/an+I * ... * 6k ))^ (Y) _
µ((Ak An
1)y)
n - -.
Consequently, has limit µ(y)/µ(exp(- tQ*) y) as `Qk(µ'n+I * ... * µk) - k L, so µ, is infinitely divisible [cf. Proposition Q.E.D. 1.7.6(b) and Theorem 1.8.11]. We conclude this section by introducing the notion of a U-exponent of a probability measure. Namely, we say that Q E End is a U-exponent of a measure µ provided its Urbanik decomposability semigroup D(µ) contains the one-parameter semigroup (e-`Q: t > 0). By Theorem 3.3.5, we see that full operator-selfdecomposable measures have U-exponents with the property that lim, __ a-`Q = 0. However, we do
not require this condition in the definition of U-exponents. Also,
fullness of µ is not necessary for some properties discussed below. Let 0. (b) 8u(µ) is closed in the operator norm topology.
(c) 8u(µ) + 8u(µ) = 8,,(14). (d) ForA E A0(µ) A(µ) n Aut, A8u(µ)A-' = 8,,(µ).
(e) 8u(µ) n (- 8u(µ)) = 7(A(µ)) n (- .T(A(µ))), and this is the largest linear subspace contained in 8u(µ), while 8u(µ) - 8,,(µ) is the smallest linear space containing 8u(µ).
Proof (a) Since the identity is in D(µ), 0 is always a U-exponent of any µ. t/a and v,. From a-`12µ * µ, = e-"("Q)µ * v,. where t'
µ
we obtain 0. (b) Let Q E 8u(µ) for all n >- 1 and assume Q, -* Q in End. Let v,, be such that µ = e``Q^µ * v, . By the continuous mapping theorem (cf. Theorem 1.6.4), e-`Q^µ e-`Qµ for all t > 0. This convergence and Theorem 1.7.1(b) imply that (v,,,,: n 2: 1) v, (some) whenis conditionally compact in .9'. [in fact, v,,,,
ever the Fourier transform µ never vanishes as when µ is infinitely divisible; cf. Proposition 1.7.6(b).]
Hence, µ =
e-`Qµ * v, for any limit point v, of the set (v,,,,: n >- 1), that is,
Q e 6"'(µ) (c) Let Q1 , Q2 E 8u(µ). Then e-`Q'e`Q2 E D(µ) for all t >- 0 and t-1[e-'Q,e-tQ2
- I] --> _(Q1 + Q2)
as t - 0. By Lemma 1.5.1, we see that 8u(µ)
Y_(D(µ)), so
Q1 + Q2 E 8u(µ), that is, 8u(µ) + 8u(µ) c 8u(µ). The converse inclusion follows from the fact that OZ 8u(µ).
(d) For A E A(µ) n Aut, µ = Aµ * S(a) for some a, and for Q E 8(µ), µ = e`Qµ * µ, for some µ,. Hence, µ = e`QAµ * µ, * S(e`Qa). Together with µ = A-'µ * 3(-A -'a) we have
µ = A-'e-tQAp * A-Iµ, *
5(A-'e-tQa -A-'a).
Therefore, A-'QA E 8u(µ), that is, A-18,,(µ)A c 8u(µ). The converse inclusion follows from A(µ) n Aut being a group (cf. Proposition 2.3.4).
URBANIK SEMIGROUP OPERATOR-SELFDECOMPOSABLE MEASURE
87
(e) From (a) and (c), we have that 6u(µ) n (- u(µ)) is a linear subspace of Clearly, it is the largest subspace of Similarly, 6u(µ) - d' (µ) is the smallest linear space containing d" (µ). The inclusion 57(A(µ))
(- u(µ))
n (- ,91(A(µ))) c eu(µ) n
follows
from Lemma
1.5.1,
because u(µ) =
- .91(D(µ)) and, of course, A(µ) C D(µ). Conversely, if Q and - Q are U-exponents, then Lemma 1.5.1 gives that, for t >: 0,
exp(-tQ) and exp(tQ) are in D(µ). From Proposition 2.3.4,
exp(±tQ) are in A(µ), so Q E 9-(A(µ)) n (-,91(A(µ))). Q.E.D.
Corollary 3.3.7.
If µ is full on Rd, then
(a) _ 0 and 0 < det(e-'Q) < 1. Therefore, Q E
9"(D(µ) n (A: 0 < det(A) < 1)), which establishes (a). (b) Since A(µ) is a compact group in Aut (cf. Corollary 2.3.2), H exists. By Proposition 1.6.8, we see that Q,, is the limit of a sequence of the form EakgkQgk', with ak >_ 0, Eak = 1, and gk E A(µ). [Note that the functions from A(µ) to R' given by g H (gQg-'x, y) for x, y fixed are continuous and bounded.] Consequently, (a), (b), and (c) of Proposition 3.3.6 imply that Q, E 9''(A). Commutativity of Q, with A(µ) is a simple conseQ.E.D. quence of the invariance of the Haar measure. Let d (µ) denote the set of all U-exponents which commute with the symmetry semigroup A(µ), and let
DJµ) :_ (A E D(µ): AB = BA for all B E A(µ)). Clearly, DJµ) is a closed subsemigroup of D(µ) and I (=- Djµ). Corollary 3.3.9.
Let µ be full on 08 d. Then
ifcu(µ) = ,T(Djµ) n (A: 0 < det A - 0. Differentiation with respect to t gives e-`Q(QS + SQ*)e-`Q* >- 0
for all t Z 0. In particular, QS + SQ* >_ 0. Conversely, if
QS+SQ*>-0,then (e-`Q(QS + SQ* )e-`Q*x, x) >- 0
for all t >- 0 and for all x c- Rd. Equivalently,
d [(Sx, x) - (e-`QSe-`Q*x, x)] >- 0, and so S - e-QSe-`Q* >- 0 for all and Q E 4u(y).
t z 0. Therefore, a-`Q ED(Y)
Since (b) is a consequence of (e) in Proposition 3.3.6, and (c) follows from (b), the proof is complete. Q.E.D. CHARACTERISTIC FUNCTIONALS OF exp(-tQ) -DECOMPOSABLE MEASURES 3.4
In this section, we specialize the concept of operator-selfdecomposable to a particular Q with exp(-tQ) -* 0 as t --> X. The assumption
of fullness in Theorem 3.3.5 was used only in the proof of the necessity, so we may drop that assumption in this section. For Q with
exp(-tQ) -> 0 as t - oo, we say that a measure µ is exp(-tQ)decomposable if for each t >_ 0, there is a measure y, such that
,a = exp(-tQ)µ * vt.
(3.4.1)
From the last part of Theorem 3.3.5, each v, is infinitely divisible. The
OPERATOR-SELFDECOMPOSABLE MEASURES
90
assumption on Q implies v, - µ as t
so µ is also infinitely
divisible [cf. Proposition 1.8.1(b)].
For an infinitely divisible µ = [a, R, M] to be exp(- tQ)-decomposable, there must be some compatibility conditions between Q and
R, and between Q and M. If µ = [a, R, M] and A E End, then (AA)^ (y) = exp i(y, Aa) -
+ f exp i(y, Ax) - 1
1
2
(ARA*y, y)
11 y, Ax) 1+11x112
M(dx) = [a, ARA*, AM],
where a
Aa +
f
IlxllZ (1
-
IIAX112
+ IIAx112)(1 + 11XII2)
AxM (dx )
and the integral exists because the Levy measure M has the property that IIIx112/(1 + II xII2)M(dx) < oo. To obtain the parameters of the convolution of two infinitely divisible measures, one simply adds the individual parameters. Consequently, by Proposition 1.8.9, (3.4.1) is equivalent to the following proposition. Proposition 3.4.1. A measure µ = [a, R, M] is exp(-tQ)-decomposable if and only if
(a) for each t > 0 and each y E Rd,
(Ry,y) >(exp(-tQ)Rexp(-tQ*)y, y); (b) for each t >- 0 and each Borel subset B of Rd \ {0},
M(B) > M(exp(tQ)B). In order to characterize exp(-tQ)-decomposable measures, in terms of characteristic functions, we shall solve these inequalities. Theorem 3.4.2. Let Q be an operator with exp(-tQ) - 0 as t - oo. A Gaussian covariance operator R satisfies the inequality
R > exp( - tQ) R exp( - tQ*)
FUNCTIONALS OF cxp(-IQ)-DECOMPOSABLE MEASURES
91
for all t > 0 if and only if
R = f exp(-tQ)T exp(-tQ*) dt,
(3.4.2)
0
where T is a unique Gaussian covariance operator; in which case,
T=QR+RQ*. Proof. Assume R is of the form (3.4.2) and fix y in l°. Then, for
t>-0, (exp(-tQ)Rexp(-tQ*)y, y) (exp(-sQ)T exp(-sQ*)exp(-tQ*)y,exp(-tQ*)y) ds
= f0
= f (exp(-(t + s)Q)T exp(-(t + s)Q*)y, y) ds
s (Ry, y). Thus, exp(-tQ)R exp(-tQ*) 5 R. Now, assume that R satisfies the inequality. Define T to be QR + RQ*. We will show that R and T satisfy (3.4.2) and that T is unique. For fixed y, set
g(t) :=((R - exp( - tQ) R exp( - tQ*)) y, y)
fortz0.Then g(0)=0,g(t)>_0forallt>0,andg(t)->(Ry,y) as t -> oo. Also, g is differentiable and
g'(t) =(exp(-tQ)(QR + RQ*)exp(-tQ*)y, y). Hence,
g(t) = f (exp(-sQ)T exp(-sQ* )y, y) ds + constant. Since g(0) = 0, the constant must be zero. Letting t - oo, we obtain
f (exp(-sQ)T exp(-sQ*)y, y)ds = Ry,y), 0 establishing (3.4.2). Clearly, T = QR + RQ* is a symmetric operator.
OPERATOR-SELFDECOMPOSABLE MEASURES
92
To see that T is a Gaussian covariance operator, it suffices to show that (Ty, y) >- 0 for all y (cf. Theorem 1.8.6). Suppose to the contrary that (Tyo, yo) < 0 for some vector yo. By continuity, there is to > 0 such that, for all s in [0, to],
(exp(-sQ)Texp(-sQ*)yo, yo) < 0. Using this yo to define the function g above, we have that g(to) < 0. However, g(t) z 0 for all t z 0. This contradiction establishes that T is a Gaussian covariance operator. For R given by (3.4.2), we obtain
QR+RQ*= f(-exp(-tQ))'Texp(-tQ*)dt w + fo exp(-tQ)T exp(-tQ*)Q* dt
_ -exp(-tQ)T exp(-tQ*)I'=o = T and hence we also infer that T is unique in (3.4.2).
Q.E.D.
Before solving the second inequality in Proposition 3.4.1, we introduce a new norm v Ib associated with the one-parameter semigroup (exp(-tQ): t > 0). We always assume exp(-tQ) -, 0 as t -> oo. This new norm leads to the natural polar coordinates for R . In these polar coordinates, orbits of the form IItQyIIQ, t > 0, are strictly increasing, so they hit the new unit sphere exactly once. For x in 118°, define IIxIIQ by
IIxIIQ =
f ll exp(-tQ)x1I dt = o
f'IIsQxJIs-1 dz.
(3.4.3)
0
Since Ilexp(-tQ)II < a exp(-bt) for some positive constants a and b, the integral is finite. Clearly, II IIQ is a norm. Also, IIxIIQ -
- IIxII f0 -IfeXp(tQ)II-1 dt.
FUNCTIONALS OF exp(-tQ)-DECOMPOSABLE MEASURES
93
Thus, for all x,
(J°°IlexP(tQ)lr' dt)llxll s IIxIIQ _5 bIIxII. 0
Let S. be the unit sphere in the norm II IIQ, that is, SQ = (z E Rd: IIzIIQ = 1).
Define (DQ: SQ X R -4 Rd \ (0) by (DQ(z, t)
tQz
(the index Q will be omitted if Q is fixed). Proposition 3.4.3. The function (D is a homeomorphism and, for fixed x in Rd \ (0), the function t
IItQxIIQ
for t > 0, is strictly increasing.
Proof. For t > 0 and x in Rd \ (0), the equality IItQxIIQ = f tllsQxlls-' ds 0
shows that (d/dt)IItQxIIQ > 0, so the functions t -> IItQxIIQ, x * 0, are strictly increasing. Furthermore, since IltQll - 0 as t -> 0, from IIt-Qxll, we get the inequality Ilxll oo
as t-400.
Thus, the orbits (tax, t z 0), for x # 0, intersect SQ exactly once. Also, 4) is one-to-one and onto. Clearly, it is continuous. Now, we show that I is also continuous. Let z,, -' z0 and let t,, > 0 and x in SQ be such that z = tQx for n >_ 0. We show that (x, (x0, t0). Suppose
contains a subsequence which converges to zero.
Then z0 would be zero. Hence, is bounded away from zero. Similarly, (t,,) is bounded above. Let t be any limit point of the set
OPERATOR-SELFDECOMPOSABLE MEASURES
94
--+
x,' = tn,Qze -p t-Q(t0 xo)
Hence, II(t/to)-QxoII Q = 1 and xo E SQ. Thus, t/to = 1, that is, t = to.
Consequently, t,, - to > 0. This implies x,, -+ xo in SQ. Therefore, Q.E.D. 4)-' is continuous and P is homeomorphism. Later, we use the following notation for B and C, Borel subsets of SQ and QB+, respectively,
[B,C]
1(B x C) =(tax:x(=- Band tEC).
By the previous proposition, for s > 0,
sQ[B,C] = [B,sC]. Let µ = [a, R, MI. Define its Q-Levy spectral function LM (for simplicity we suppress the index Q) as follows. For r > 0 and B a Borel subset of SQ,
LM(B;r) __ -M(1(B x [r,°°))) = -M([B,[r,oo)]) Since M is a finite measure outside every neighborhood of the origin, Proposition 3.4.3 implies that -LM(-; r) is a finite measure on SQ for
each r > 0. Furthermore, LM uniquely describes M. This Q-Levy spectral function determines whether [a, 0, M ] is exp(- tQ)-decomposable.
Theorem 3.4.4. A Levy measure M satisfies the inequality M >exp(-tQ)M for all t >- 0 if and only if, for each Bore! subset B of SQ, its Q-Levy spectral function LM(B; - ) has both right and left derivatives everywhere on (0, oo) and the function a
r -* r-LM(B; r) ar
is nonincreasing. Here, the derivative denotes either the right or left derivative, possibly changing with r or changing from left to right at the same r.
FUNCTIONALS OF exp(-IQ)-DECOMPOSABLE MEASURES
95
Proof. Assume that r(a/ar)LM(B; r) is nonincreasing. Then (a/ar)LM(B; r) is also nonincreasing. Hence, LM(B; ) is continuous and concave. Furthermore, for h E (0, 1) and r > 0, a
arLM(B; r)
>
1a
r
h arLM(B; h )
Hence, for 0 < a < b,
a
M([B; [a, b)]) = f (a, b)
ar 1
Lm
r) dr
a
r
> f[a, b)--LM(B; -) dr h ar h
[a/h, b/y) ar
LM(B; r) dr
= hQM([B, [a, b)]). Since 1 is a homeomorphism, M(A) >- hQM(A) for all Borel subsets A of Rd \ (0). This establishes the sufficiency. Now, assume that M >_ exp(-IQ)M for all t >- 0. Suppose there is to > 0 such that M([B; {t0}]) > 0. Then, for h in (0, 1), M([b, {hto}]) >_ hQM([B, {hto}]) = M([ B, {to}]) > 0,
which is impossible. Hence, LM(B; ) is continuous. For t < s and h > 0, LM(B; e') - LM(B; es)
e-hQM([B, [e`, es)])
= LM(B; el+n) - LM(B; es+b). Hence, t --+ LM(B; e`) is continuous and concave on W. The wellknown properties of such functions yield the condition. Theorem 3.4.5.
A Levy measure M satisfies the inequality
M > exp(-tQ)M
Q.E.D.
OPERATOR-SELFDECOMPOSABLE MEASURES
96
for all t >- 0 if and only if there exists a unique Levy measure G with log(1 + IIxII)G(dx) < co
f
IIXII> I
and
M(B) = f G(e'QB) ds 0
for any Borel subset B of Rd \ (0).
Proof. Assume M and G are related as in the theorem. Then
exp(-tQ)M(B) =
WG(exp((s + t)Q)B) ds < M(B). f0
Now, assume M z exp(-tQ)M for all t >- 0. Let LM be the Q-Levy spectral function of M given by (3.4.3). Define g and j on SQ X R
as follows: a
g(A, t) `= at LM(A; t), (A, t) tg(A, t),
where the partial derivative denotes the left derivative which is left-continuous. By Theorem 3.4.4, the functions g(A, ) and g(A, ) are nonincreasing and left-continuous. Also,
M([A,[t,oo)] ) = -LM(A;t) = f g(A, s) ds = f s-'g(A, s) ds. t
(3.4.4)
t
Since M is finite outside every neighborhood of the origin, these integrals are finite, so
lim g(A, t) = 0 = lim g(A, t).
t-W
Therefore, for each A in O(SQ), there exists a unique Borel measure
FUNCTIONALS OF exp(-,Q)-DECOMPOSABLE MEASURES
97
on R+ such that, for 0 < a < b,
PA([s, b)) `= g(A, a) - g(A, b). In particular, AA a, oo)) < oo for all a > 0. We now show that for fixed 0 < a < b, the function
A -' AA([ a, b)) is a finite Borel measure on q(SQ). By the definitions of LM and we see that µA([a, b)) is finitely additive in A. Hence, it suffices to show that it is continuous at the empty set, that is, for every decreasing sequence {An} in Q(SQ) which converges to the empty set, we have µA([a, b)) - 0. This would follow from g(An, t) -' 0 for each
t > 0. Suppose to the contrary that there exist t > 0, s > 0, and sequence (An} in 9(SQ) decreasing to the empty set such that g(An; t) >- e for all n >_ 1. Then, for all n >_ 1,
M([ An, [t/2, t)]) = f' g(An, s) ds >- et/2 > 0, since g(A, ) is nonincreasing. But, M([An, [t/2, t)]) 0. Therefore, for fixed 0 < a < b, µA([a, b)) is a finite Borel measure on 9(SQ). We now extend µA(I), A E 9(SQ), I = [a, b) c UB+, 0 < a < b < oo, to finite Borel measure on 9(SQ x N, oo)), where 77 > 0 is arbitrary. Let
R1,:=(AXI:AE. 9(SQ), I
[a, b)9;R+ for00, M((D(A X [t,cc))) = f g(A, es) ds = f g(A, tes) ds. log r
0
Consequently, for A E . (SQ) and B = It, u) with 0 < t, w M([ A, B]) = fPA(e'[t, u)) dr
= f p(A X e'[t, u)) dr =
fG(4)(A X e'[t, u))) dr
W
=f G(e'Q[A, B]) dr. 0
By Proposition 3.4.3, this last equality holds for arbitrary F E Q(Rd \ {0}).
Next, we show that fuXu,1 log(1 + II xI DG(dx) < c. Let F. [SQ, (1, co)], f(t) := G([SQ, (t, oo]]), and f(t) := G({x: IIxIIQ > t}) for t > 0. Then M(F0) < cc. For t > 1, we have IItQII s t"Q11, so for t > 1, letting y := 1/IIQIIQ, {IIxIIQ > t} = {sQz: IISQZIIQ > t, s > 1, z e SQ},
{sQz: s > t'', z E SQ} = [SQ,(t7,co)].
OPERATOR-SELFDECOMPOSABLE MEASURES
100
Consequently, for t > 1,
f(t) M(F0) = JOOG(e'QFo) dr = f wG(sQF0)s-' ds =
f
0
f(s)s-' ds = y
f
f(uI)u-' du.
Thus,
logll xFIQG(dx) = f f (s)s-' ds < f f(sY)s-' ds < oo. 'DXIIQ> 1
1
1
Hence, f lIXII, >I log(1 + ll x l (Q)G(dx) < oo. Since the norms II
II
Il and
IIQ are equivalent, we have fIIXII>, log(1 + IIxII)G(dx) < oo.
To prove that G is a Levy measure, we see 00 > f
Nx112M(dx) = f f
O 0 and x E Rd (3.4.6)
1
OPERATOR-SELFDECOMPOSABLE MEASURES
102
and
r c51og(1 + 11x112)
-
Ile-tQxll2 1 + Ile-`Qx112
< c6 log(1 + 11x112)
dt
for all x r= Pd. (3.4.7)
Proof. Let a1, ... , aP denote the eigenvalues of Q and let
0 < c4 < min Re(al) 5 max Re(al) < c2 < 00. 15j5p
1515p
Then
lim e`4t exp(-tQ) = 0,
1-00
lira e`2` exp(-tQ) = 0. r-.-W
Hence, C3 := sup rz0
ci
IIe`41 eXp(-tQ)II,
sup II e`2t eXp(-tQ) II
rs0
are positive and finite. Consequently, for all t > 0 and all x e UBd, IIeXp(-tQ)xll < c3e "'Ijx11. Also, for t > 0 and x E=- R e-`2`11x11 0, there is a constant K2 such that (3.4.11) is bounded by K2 log(' + IIx112) for x c- Rd with IIxII >: c31. Letting max(Kl, K2), we obtain the desired bound on Ilh(x)11 for all x in K Q.E.D. Rd, which completes the proof. From the integral representation of the Levy measures corresponding to the exp(- tQ)-decomposable measures (cf. Theorem 3.4.5), we obtain the following corollaries. Corollary 3.4.9. Let M and G be positive Bore! measures on Rd vanishing at {0} and such that
M(F) = f G(e"QF) dt for F E 9(Q8d \ (0)), 0
where Q is a fixed invertible linear operator. Then
(a) For any A E A1 d) such that exp(tQ)(A) = A for all t >- 0, we have that A and R d \A have either zero or infinite M-measure; so, in particular, for any Q-invariant subspace V of Rd, either and either M(P d \ V) = 0 or M(V) = 0 or M(V)
M(48d\V)=00; (b) M([SQ, CI) = 0 for any countable subset C in R+; in particular, M has no atoms. Proof. (a) is obvious, and (b) follows from the following equality: M(ASQ,CDD = J
fl[sQ,cj(e-SQx)dsG(dx)
Rd\{o) o
=
f
11({s: e-sQx E ESQ,C])G(dx) = 0,
where 1, denotes the Lebesgue measure on R1.
Q.E.D.
For an arbitrary subset A of Pd, let lin(A) denote the smallest linear subspace of Rd and linQ(A) the smallest Q-invariant subspace of 18d containing the set A, respectively.
Proposition 3.4.10. For positive measures M and G related as in Corollary 3.4.9, we have
(a) supp M = W, z oe-tQ(supp G))-, where A -denotes the closure of a set A; (b) lin(supp M) = linQ(supp G).
OPERATOR-SELFDECOMPOSABLE MEASURES
106
Proof z0e-'Q(supp G)) -. Then there is (a) Let x E supp M and x t (U an open set U containing x and U n (e-tQ(supp G)) = 0 for each t 0. Hence e'QU is open and disjoint with supp G, for each t z 0. Therefore,
M(U) = f G(eQU) dt = 0, 0
which U
contradicts x e supp M. Conversely, let x C120(e-'Q(supp G)). Then x = e-'°Qgo for some to > 0 and
supp G. For each open set U containing x, there are an open interval (to - 3, to + 3) and an open set V containing go such that e-'Qg E U for all t E (to - S, to + 5) and all g E V. Since etQU is also open and go E e'QU, for t E (to, to + S) we
go
get
M(U) > ft0+SG(e'QU) dt > 0, to
that is, x E supp M. Consequently, (U 1> o(e-tQ(supp G)))-c supp M which completes the proof of (a).
(b) Let K := U 0e-'Q(supp G). Clearly, a-SQx E K, whenever x c- K and s 0. Since Qx = limit of (x - e-'Qx)/t as t 10, we have Qx E lin(K) for any x e K, that is, lin(K) is a Qinvariant subspace containing supp G. Hence, linQ(supp G) c lin(supp M). Conversely, since Qc(supp G) c linQ(supp G) for k = ... , we have e-SQ(supp G) c- linQ(supp G) for all s >- 0. Finally, 0, 1, 2,
lint U (e-SQ(supp G))
kto
c lin(linQ(supp G)) J
= linQ(supp G), which completes the proof.
Q.E.D.
Remark 3.4.11. If a-tQ - 0 as t - oo and supp G is also bounded in 08 d, and hence compact, then
( U e-tQ(supp G)) = U e-tQ(supp G) U tzO
ta0
(0).
FUNCTIONALS OF cxp(-rQ)-DECOMPOSABLE MEASURES
107
Remark 3.4.12. From Theorem 3.3.5, we see that the formulas (3.4.5) and (3.4.8) as well as Theorem 3.4.5 and all the above corollaries characterize full operator-selfdecomposable measures. Proposition 3.4.13. tors related by
Let R and T be nonnegative self-adjoint opera-
R = Io
e-`QTe-`Q* dt
0
(cf. Theorem 3.4.2). Then we have
(a) ker R= n, a oe`Q*(ker T), and ker R is a Q*-invariant subspace;
(b) R(Rd) = linQ(T(Rd)).
Proof (a) For a nonnegative selfadjoint operator D, we have J (Dx, Y) 12 < (Dx, x) (Dy, y) by the Schwarz inequality. Hence, ker D = {x: (Dx, x) = 0). Therefore, x e ker R if and only if
jo(e-`QTe-`Q*x, x) dt = 0 if and only if (e-QTe-`Q x, x) = 0 for all t >_ 0 if only if x E ker(e-`QTe-`Q*) = ker(Te-`Q*) = e`Q*(ker T) for all t >- 0. Thus, ker R= n,., oe'Q*(ker T). To see that ker R is Q*-invariant, note that e-'Q*(ker R) c ker R for all s > 0. Hence, for any x e kerR, (x - e-SQ*x)/s c= ker R. Consequently, Q*x = limslo(x - e-'Q*x)/s E ker R whenever x E ker R. (b) In the following, we use the facts that (V, n V2)1 D V,1 U V21 and (A*(V,))1= A-1(V,1), for any subspaces V, and V2, and
for any invertible linear operator A. Note that R(R) _ (ker R)1 and T(OR") = (ker T)1 since R and T are selfadjoint. Using (a), we have R(Rd)
(fletQ*(kerT))1D U (e`Q*(ker T))1 ta0
t- U
= U e-'QT(Pd) ra0
Hence, R(6Rd) D T(QRd). Since ker R is Q*-invariant, we have R(Rd) is Q-invariant. Therefore, R(W') Q linQ(T(6Rd)).
Now, for any x E W', e-`QTe-`Q x E linQ(T(Pd)) for all t z 0. Hence, Rx = joe-`QTe-`Q*xdt c= linQ(T(ORd)), that is, R(Pd) S linQ(T(Rd)). Q.E.D.
OPERATOR-SELFDECOMPOSABLE MEASURES
108
3.5 GENERATORS OF THE CLASS L0(Q) We assume the linear operator Q on 68d is such that exp( - tQ) -* 0 as
t --> oo in the norm topology. By L0(Q) we denote the class of all exp(- tQ)-decomposable measures, that is,
L0(Q) :_ {µ E 9: for each t > 0, there is v: E such that µ = e-'Qµ * vt}, (3.5.1)
that is, µ E L0(Q) if and only if its Urbanik decomposability semigroup D(µ) contains the one-parameter semigroup (e-'Q: t E R'). Note that for c > 0, L0(cQ) = L0(Q). Thus, without loss of generality, we may assume that IItQII rar LM(A, r) is nonincreasing on R', for A E `(SQ).
OPERATOR-SELFDECOMPOSABLE MEASURES
110
The simplest nonincreasing functions are given by
0 < r -+ m(A)I(o,al(r), where m(A) > 0, a > 0.
(3.5.4)
For such functions,
M([A, [r,oo)]) = m(A) f WI(Q,aj(t)t-1 dt = m(A) logmax(1, r
a r
),
must be a finite measure on R(SQ). Finally, let
and therefore
MQm(F) :=sf fo IF(sQx)s-1 dsm(dx),
FE
(3.5.5)
Q
For ease of notation, we omit the index Q and simply write Ma,m. Note that Ma,,,,(F) < oo for every F separated from zero, that is, 0 F, and
f
1SQ,(0,111
II xIIMa,m(dx) =SQf of
aIIsQxIIs-i dsm(dx)
0 and h is nonincreasing, we obtain
f(c) -f(a) > h(d) > h(e) > f(b) -f(c)
g(c) - g(a)
g(b) - g(c)
Q.E.D.
OPERATOR-SELFDECOMPOSABLE MEASURES
112
Lemma 3.5.4. For each µ = [0, 0, M] belonging to L0(Q), there exist a subsequence (kn) of positive integers, positive real numbers anj, and finite Borel measures mn1 on SQ, 1 < j 5 kn, 1 0 with i e 12,3,..., kn) such that (i - 1)/2"
2, we define the functions Lni(F, r) as follows:
Lni(F, r)
i-1
2"r ani(F)log(
i
)
+ bni(F),
2"
0, there exists a positive integer no such that, for all n >- no and all k = [2"r] + 1, ... , kn, we have rM (F,
k
/
2n") - LM I F,
k-1 2n
-LM(F, n) < E.
) < E,
Therefore, H"(F, r) -. LM(F, r) as n --+ oo, which establishes (3.5.10). Now, we prove (3.5.11). In view of (3.5.7), (3.5.8), and (3.5.9) and using the partial summation formula,
f
IIxII2Nn(dx) = [SO,(0.ejl
e
f IltQxll2t-' dtmnk(dx)
k-1 SQ 0
= f f 2 niItQxII2t-' dtan2(dx) SQ 0
k"
f
r J
KlL..
+ k=2 SQ (k-1)/2"IItQXII2t-' dtank(dx) e
+
II
I1tQxII2t-' dtQn,[2"e]+1(),
SQ [2"el/2n
(3.5.12)
where [b] denotes the greatest integer less than or equal to b. We consider the second term in (3.5.12). By the first mean value theorem and by (3.5.3), for x e SQ and for the interval Ink ((k -
1)/2", k/2"], with k = 2,...,[2'--], there exists tnk in Ink such that k
f IItQxII2 d(log t) = Ilth XII2 logs k 'nk
k-1 -g2(
2"
1/2 [cf. (3.5.2)]. This bound goes to zero as n -> oo for each e > 0. Thus, (3.4.11) is established. Q.E.D. From Proposition 3.5.1 and Lemmas 3.5.2 and 3.5.4, we obtain the main result of this section.
OPERATOR-SELFDECOMPOSABLE MEASURES
116
Theorem 3.5.5. The class LO(Q) is the smallest closed subsemigroup of ID containing the generators . 'Q.
Remark 3.5.6. The proof of Lemma 3.5.4 used the assumption (3.5.2) with y > 1/2. However, Theorem 3.5.5 holds for any Q with
To establish this remark, we need only consider the relationship between .7d'Q and XQ for a > 0. Since IIxIIaQ = f oIle-b0QxII dt = a-'IIxlIQ, 0
we have SaQ = aSQ.
Hence, if m is a finite Borel measure on SaQ, then
a-'m(aA),
ma(A)
A E 9(SQ),
is one also on SQ. Note that c(DQ(t, u) = 4)cQ(t'1 `, cu).
Thus, we have MaQ
(F) =
saQ
f
IF(saQx)s-' dsm(dx)
= a-' f f IF(tQay)t-' dtm(ady) SQ 0
= MQ,..(a-'F) for F E ARd \ {0}), and this establishes the remark. 3.6 RANDOM INTEGRAL REPRESENTATION
Equation (3.4.1) can be written in terms of random elements as follows: "An fed-valued r.v. X is exp(-tQ)-decomposable if for each
117
RANDOM INTEGRAL REPRESENTATION
t >- 0 there exists an X, independent of X such that
X=e-`QX+X
(3.6.1)
where = means equality in distribution and r.v. means random variable, random vector, or synonymously random element." Since we
assume a-`Q -> 0 as t --* - and since for each a > 0 we also have X = e-`IaQ)X + Xat,
we may assume without loss of generality that He-Q11 < 1,
(3.6.2)
by replacing Q with aQ for sufficiently large a > 0. Moreover, from (3.4.1) and its following statement, we see that X is infinitely divisible. Also, (3.6.1) shows that X is the limit in distribution of the stochastic The main aim of this section is to prove process (Xt: t >- 0) as t d that X, Z where Zt has a precise random integral form. For this purpose, we define the necessary random integrals using the device of formal integration by parts even though our integrand is a deterministic one-parameter operator semigroup. Namely, for the one-parameter semigroup (e-Q: t > 0) and for the D(IIBd, [0, co))-valued r.v. Y, we set
'eca -sQ bl dY(s) := e-bQY(b) - e-aQY(a) - fabld(e-'Q)Y(s) J(a b) ds = e-bQY(b) - e-'QY(a) + /'Qe-SQY(s)
(3.6.3)
for 0 < a < b < oo, where D(OBd, [0, cz))-valued means the Skorohod space of all functions from [0, oo) to Rd which are right-continuous and have finite left limits (cf. Section 1.9). The last integral in (3.6.3) exists because of the following property of D(OB", [a, b]). Lemma 3.6.1. For each y in D(OBd, [a, b]) and each e > 0, there exist points to, ... , tr such that
a =:to< ... -, the limit is denoted by joe-tQ dY(t) and foe -'Q dY(t) [a", R°°, M'], where
R°° = f -e-`QRe-`Q* dt,
(3.6.9)
0
M°°(F) = f M(etQF) dt for F E .(Rd \ (0)),
(3.6.10)
0
a°° = f we-`Q(a + bM(t)) dt 0
= Q-'a + f e-tQbM(t) o dt.
(3.6.11)
0
Our immediate goal is to describe when the convergence in question holds. We start with the following auxiliary lemma on random operator-power series.
RANDOM INTEGRAL REPRESENTATION
121
Lemma 3.6.5. Let A be an invertible linear operator with II All < 1 and let {in) be a sequence of independently distributed r.v. Then W
F, A'
converges a. s.
n=1
if and only if E log(1 + 116111) < oo.
Proof. Let c
II A II Since A is invertible, we have s IIA"6nll s c"116,,11
for all n >_ 1. Assume converges a.s. From Kolomogorov's three-series theorem and the above inequality, we have
dllA-1II"]
0. Consequently, E P [ log it 11l > n log f I A `III < 00, n
which is equivalent to E log+116111 < -, so the necessity is established. Conversely, if E then, for each d E (1,1/c),
F,P[log+Ilf1II >_ n log d] = FP[llfllll" >_ d] n
n
= EP[IIC"enll11" > Cd] < 00. n
The Borel-Cantelli lemma implies that Pf Jimsupllc"fnII1/" LL
n-w
>_
cdl = 0. JJ
Since cd < 1, we see that EnA"fn converges a.s.
Q.E.D.
Now, we can give the complete characterization of any D(68d, [0, oo))-valued r.v. Y with stationary independent increments such that foe `Q dY(t) exists.
122
OPERATOR-SELFDECOMPOSABLE MEASURES
Theorem 3.6.6. Let {exp(- tQ): t > 0} be a one parameter semigroup with lim, __ e-`Q = 0. Let Y be a D(R', [0, c))-valued r.v. with
stationary independent increments, Y(0) = 0 a.s., and J"(Y(l)) _ [a, R, M]. Then the following statements are equivalent.
(i) foe -'Q dY(t) converges in distribution as s - oo.
(ii) foe-`Q dY(t) converges in norm a.s. ass - 0. (iii) E log(I + II foe-tQ dY(t)ID < oo for given s > 0. (iv) E log(1 + IIY(1)ID < -. (v) fuxu>1 log(1 + IIxIDM(dx) < oo.
Proof. In view of (3.6.2) we may assume IIe-Q11 < 1 (cf. also Remark 3.6.7). (iv) (ii) (i)
(v) This is shown in Proposition 1.8.13. (i) This implication is obvious.
(ii) Note that if 0 = so < s < s,, +1 -* oo, then n
Z(sn) =
40, Sn1
e-`QdY(t) _ E f
e-`QdY(t)
k=1 (Sk_I,Sk]
and the summands are independent. Therefore, the convergence in distribution of {Z(sn)) is equivalent to a.s. convergence. Since the sample paths of Z are right-continuous with left limits, the exceptional null set does not depend on the particular sequence (sn) tending to infinity (cf. Remark 1.9.11).
(i) - (v) Let µ be the limit measure in W. From (3.6.10), we see that the Levy measure of µ has the form
f M(e`Q ) dt. 0
Theorem 3.4.5 gives that M has a logarithmic moment on every open complement of zero. (v) - (iii) By Proposition 1.8.13 and (3.6.7), it suffices to show that logllxllMs(dx) < oo Ilxll> 1
for all s > 0. There is a > 0 such that for all t > 0, He-`QII < a.
RANDOM INTEGRAL REPRESENTATION
123
Hence,
I
logll xII M8'(dx)
1411> 1 S
=f0 ' {x : Ile-°Qxll> 1} loglle-`QxIIM(dx) dt
s sf
log all xlI M(dx)
(x: ailxR> 1)
= sf
(x:Ilxll>a ')
logllxllM(dx)+sM(Ilxll >a-')log a.
The last integral above is finite by (v) and the remaining term is finite since a Levy measure puts finite mass on any set bounded away from zero. (i) Note that (iii)
f
n-1
e-`Q dY(t) = F, e-kQ f k=0
(0,n)
e-`Q dY(t + k).
(0,1)
Let
k=f
e-`Q dY(t + k).
(0,11
Then
e`Q dY(t), k - f (0,1]
k = ekQ f
e `Q dY(t).
(k, k+ 1)
The equality in distribution is due to the stationarity and independence of the increments of Y (cf. Lemma 3.6.4). Hence, are also independent identically distributed random variables (cf. Lemma 3.6.3). By our assumption (iii), each CA: has a finite logarithmic moment. Hence, by Lemma 3.6.5,
f
e-`Q dY(t) - W a.s. as n
(3.6.12)
(0, n]
for some random variable W with -AW) = [a", R°°, M]. Furthermore, since the functions t --> -Z( J(o,,]e-SQ dY(s)) are right-continuous with left limits in .9P, we see that the set of measures {[a`, R', M`]:
OPERATOR-SELFDECOMPOSABLE MEASURES
124
0 < t < 1) is conditionally compact in .9. Hence, we have
f(n, n +tje-SQ dY(s) = e-"Qf
e-'Q dY(s) - 0
(3.6.13)
(0,
for any arbitrary sequence {tn) in (0, 11, as n - oo. Consequently, (3.6.12) and (3.6.13) imply that
f
e-SQ dY(s)
e-SQ dY(s) = f(0,[1]] e-sQ dY(s) + f([:
W
],t ]
(0,t]
as t -- oo. Thus, (iii) - (i).
Q.E.D.
Remark 3.6.7. In the proof of Theorem 3.6.6, the assumption that Ile-Q11 < 1 may be discarded because f(o.tle-SQ dY(s) converges as f(o,atle-SQ dY(s) = t -> oo if and only if, for arbitrary a > 0, f(o,t]e-s(aQ)dY(as) converges as t --+ oo; in the last integral one can
choose a > 0 such that He-'Qll < 1. Note that Y(a
)
is also a
D(Rd, [0, oo))-valued random variable with stationary independent in-
crements and _AY(a)) E ID,og for all a > 0 if and only if _Z(Y(1)) E ID,og, where ID,., is the class of all ID measures with finite logarithmic moment.
From these preliminary results we are ready to obtain the random integral representation of an exp(-tQ)-decomposable random variable mentioned at the beginning of this section. Theorem 3.6.8. Let (exp(- tQ): t > 0) be a one parameter semigroup with the property that a-tQ --> 0 as t -+ oo. Then X is an
exp(-tQ)-decomposable random variable if and only if there exists a D(Rd, [0, oo))-valued random variable Y such that Y has stationary independent increments, Y(0) = 0 a.s., and
f(0,t]e-sQ dY(s) d X as t - oo; in which case, the random variable X, in (3.6.1) satisfies
Xt = f
e-sQ dY(s)
(0,t1
for each t > 0.
(3.6.14)
125
RANDOM INTEGRAL REPRESENTATION
Proof. For the sufficiency, let X = Joe-SQ dY(s). Then, for each t > 0,
X=f
r
e-SQ dY(s) + f Joe-sQ dY(s) W
= e-IQ f e-SQ dY(s + t) + f
e-SQ dY(s).
(0,r]
0
Note that the last two terms are independent by the independent increments of Y and also fe-SQ dY(s + t)
o'e-SQ dY(s) d
X
0
by the stationarity and independence (cf. Lemma 3.6.3). Thus, X = e-'QX + X, with X and X, independent and with X, as in (3.6.14). Therefore, X is exp( - tQ)-decomposable. Furthermore, the
distribution of X, is uniquely determined since the characteristic function of X never vanishes; X is infinitely divisible (cf. also Proposition 2.3.7). Therefore, (3.6.14) must hold. Now, for the necessity, assume X is exp(- tQ)-decomposable. We
claim that this implies that there exists an 1"-valued process (Z,: t > 0) with independent increments such that Zo = 0 a.s. and, for
t,u>_0, Z,+u - Z,
a
e-tQX
in i1Bd,
(3.6.15)
so that, in particular,
Zr=X,
in Rd
To see this, it suffices to show that if (3.6.15) and (3.6.16) hold for two
particular values, say t and u in [0, oo), and Z, is independent of Zt+u - Z then (3.6.16) holds for t + u. From (3.6.1), it follows that e-(t+u)QX + Xt+u =
X
e-tQ(e-uQX + Xu) + X,
= e-((+u)QX + e-'QX + X1,
OPERATOR-SELFDECOMPOSABLE MEASURES
126
with each of X,+u, X, and X, independent of X. Since the characteristic function of X never vanishes, we have d
Xt+u
e-tQXu + X,
in 1 d.
Consequently, a-'QXu + XI d Xr+u
Zt+a = (Zt+u - Zt) + Zt
in Rd
as was to be proved. Let Z be a D(Rd, [0, oo))-valued version of the process (Z1)1Ea+ with independent increments. Now, set
Y(t) := f
esQ dZ(s)
fort > 0.
(0, t ]
From Lemma 3.6.3, we see that Y is a D(W', [0, -))-valued random variable with independent increments. Even more, Y has stationary increments. To see this, note that
Z(t + -) - Z(t)
d
e-'QZ(-) in D(Rd, [0, _)),
since both sides have independent increments and the same marginal
distributions by (3.6.15) and (3.6.16) (cf. Theorem 1.9.9). Consequently, for fixed t, u E R+, we get
Y(t + u) - Y(t) = f
esQdZ(s) = esQetQdZ(s + t) (, t+u] 40, u]
d
esQ dZ(s) = Y(u)
'(0,u]
in Rd.
Finally, from (3.6.3) and the definition of Y,
f'(0t]
e_sQesQ
e-sQ dY(s) = f
(0,t]
dZ(s) = Zr.
Hence, by (3.6.16) and (3.6.1), J
'(0,:]
eSQdY(s)
which completes the proof.
a
X,
.X
as t
-, OD,
Q.E.D.
127
RANDOM INTEGRAL REPRESENTATION
Let L0(Q) denote the set of all exp(-tQ)-decomposable measures [cf. (3.4.1) and (3.6.1)] and let ID,og denote the set of all infinitely divisible measures p. with f log(1 + II x I D p.(dx) < oo. Theorems 3.6.6 and 3.6.8 suggest the following mapping: 9Q-: IDj0g -* Lo(Q)
by means of the formula -9,Q(A)
j(
f0we-`Q
dY(t)),
(3.6.17)
where Y is a D(ll , [0, oo))-valued random variable with stationary independent increments, Y(0) = 0 a.s., and .f(Y(1)) = µ. In other words, Theorem 3.6.8 gives
Lo(Q) _ Q(IDiog) {_I'( I°°e-Q dY(t)): Y is a D(Rd, [0, oo))-valued r.v. with stationary independent increments,
Y(0) = 0 a.s., and Y(1) E ID,og).
(3.6.18)
If we set [a", R°°, M°°] := Q([a, R, M]), then a°°, R°°, and M°° are given by (3.6.9) to (3.6.11), respectively. Consequently, the equality ,70(ID10g) = L0(Q) together with Corollary 3.4.1 gives the following 0 statements. Remark 3.6.9
1. A Levy measure M satisfies the inequality M:5 e-`QM for all
t > 0 if and only if there exists a Levy measure G with the property f jjXji > I logo + II x I DG(dx) < oo such that
M(F) = j G(e`QF) dt for every FE W(Rd \ {0}). 0
2. A Gaussian covariance operator R satisfies the inequality R >_ e-`QRe-`Q` for all t > 0 if and only if there exists a Gaussian
OPERATOR-SELFDECOMPOSABLE MEASURES
128
covariance operator T such that R = f e-`QTe-`Q* o dt. 0
3. The operator T and the Levy measure G in (1) and (2) are
uniquely determined (cf. Proposition 3.6.10).
4. A measure µ is in L0(Q) if and only if there exists a unique measure v E IDlog such that
µ(y) = exp f0 Olog v(e-`Q*y) dt.
This remark shows that the random integral representation provides an alternative method of solving the operator and the Levy measure inequalities that describe the class L0(Q) (cf. Proposition 3.4.2, Theorem 3.4.5, and Corollary 3.4.8). Our next goal is to describe some algebraic and topological proper-
ties of the mapping .lQ acting between the topological semigroups L0(Q) and ID10g (cf. Proposition 3.5.1 and Remark 1.8.14 on ID,,,,). Proposition 3.6.10
(a) The mapping JQ is an algebraic isomorphism between the semigroups L0(Q) and ID,og. (b) Q(µ*S) = (, p)*S for s > 0 and p. E ID,og. (c) If the linear operator T commutes with Q, then 9Q-(TA) = T.TQ(µ) for µ E IDiog.
(d) For all s > 0, x E Rd, and µ E ID,0g, .TQ(sQp
* 3(x)) =
SQ8TQ(p) *
5(Q-lx)
Proof
-(a) Theorems 3.6.6 and 3.6.8 show that TQ maps ID10g onto L0(Q). Next, we show that 9 is one-to-one. By the definition
129
RANDOM INTEGRAL REPRESENTATION
of ,TQ and Lemma 3.6.4, we have, for all s > 0, log(.TQµ)^ (sQ*y) = f0O'log
f
-;'(Y(1))(e-`Q*sQ*y) dt
Slog
2(Y(1))(tQ*y)t-' dt.
U
Hence, for each y E Rd and s > 0, d
s(ds
lQµ)-
)log(
(sQ*y) = log
'(Y(1))(sQy)
Setting s = 1, we obtain dsd
(sQ*) y
5=1!
This implies that _,'(Y(1)) is uniquely determined by Yap,, that
is, JQ is one-to-one. To see that .lQ is a homeomorphism, let µ, v c= ID,og with = [a, R, M] and v = [b, S, N]. From (3.6.9) to (3.6.11), we see that
(M+N)M°°+N°°,
(R+S) =R°0+S°°, (a + b)=a°°+b°°.
Therefore, Y is a homeomorphism. (b) Note that µ E ID10g if and only if µ*5 E ID1og for every s > 0. From (3.6.9) to (3.6.11), we obtain the desired equality in (b). (c) Since T commutes with Q,
f
,0e
`Q dY(t )) =
_/(f
00
Te`Q dY(t))
f e-`Qd(TY(t))). Note that TY is a D(Rd, [0, c))-valued r.v. with stationary
OPERATOR-SELFDECOMPOSABLE MEASURES
130
independent increments and TY(O) = 0 a.s. Also, 2(TY(1)) E ID,og because
f log(1 + Il x11)(Tµ)(dx) < log(1 + IITII)
+ f log(1 + Ilxll)µ(dx) < oo. Hence, (c) is established.
(d) Since foe-`Qa dt = Q-'a for all a E Rd, (d) follows from (a) Q.E.D.
and (c).
Corollary 3.6.11.
Let A be invertible and µ E ID dog. Then 7AQA '(µ) = A TQ(A
consequently,
L0(AQA ') = AL0(Q) Proof. We have 7AQA '(µ)
A fo
e`Q d(AY(t)) = A.TQ(A
Since, for invertible A, Aµ E IDlog if and only if µ E ID,og, this first equality and (3.6.18) implies the second equality, that is, LO(AQA-') Q.E.D. = AL0(Q). Now, after establishing the algebraic properties of the mapping .TQ,
we wish to describe its topological properties, with the topology of weak convergence on -50. First, note that j log(1 + IIxIDy(dx) < oo if and only if f log(1 + IIXI12)y(dx) < oo for all y E .9. In particular, for the Levy measure M of a µ E ID,og, we have that, for all e > 0, 4xll>log(1 + IIx112)M(dx) < oo e
by Theorem 3.6.6. Thus, we may define, for F E 2(R d \ (0)),
M(F) := f log(1 + IIx112)M(dx). F
(3.6.19)
131
RANDOM INTEGRAL REPRESENTATION
Then M is a finite measure. The following lemma concerns the weak convergence of a sequence (Mn) of Levy measures with this integral condition and of the corresponding sequence (.4j. Lemma 3.6.12. Let M and Mn, n >_ 1, be Levy measures with finite logarithmic moments outside every neighborhood of zero and let M and Mn be given by (3.6.19). Then
M.
M outside very neighborhood of zero
and
lim sup f
log(1 + II x112)Mn(dx) = 0
s-'0D n21 Ilxll>s
if and only if Mn - M outside every neighborhood of zero.
Proof. First, we consider the "only if' part. Let f :
e 1
- R' be
continuous, bounded by K > 0, and vanishes in some neighborhood of zero. Let s > 0 be given and select s > 0 so that M({x: IIxII = s}) = 0 and sup nz1
f
log(1 + IIXII2)Mn(dx)
s
-. K E
Since f(x)log(1 + IIx1I2)I(IIxIISS)(x) is bounded and vanishes in a neigh-
borhood of zero and its set of discontinuity points has M-measure zero, we have
lim f
f(x)M,,(dx) = lim f
Ilxllss
f(x)log(1 + IIxI12)Mn(dx)
n-'°° Ilxllss
f
f(x)log(1 + IIx112)M(dx)
Ilxllss
f
f(x)M(dx).
Ilxllss
OPERATOR-SELFDECOMPOSABLE MEASURES
132
Also,
sup If nZ1
f(x)Mn(dx) < sup f
K log(1 + II xII2)Mn(dX) < e.
nz1 Ilxll>s
Ilxllzs
Hence, slim
f f(x)Mn(dx) = f f(x)M(dx),
which shows that Mn - M outside every neighborhood of zero. Now, for the converse, note that Mn - M outside every neighborhood of zero implies that the sequence (Mn) is tight. Thus,
0 = lim sup A (IIxII > s) = lira sup f S-00 nz1
log(1 + IIXI12)Mn(dx).
nz1 Ilxll>s
Furthermore, if f is continuous, bounded, and vanishes in some neighborhood of zero, so does g(x) == f(x)/log(1 + IIx112). Hence, lim ff(x)Mn(dX) = lim fg(x)Mn(dx) n- m
n
= f g(x)M(dx) = f f(x)M(dx). Therefore, Mn - M outside every neighborhood of zero.
Q.E.D.
Theorem 3.6.13. Let µn = [an, Rn, Mn] and µ = [a, R, M] belong to ID log for n >_ 1. Then
µn - µ
and
lim sup f
S-00 nz1 Ilxll>s
log(1 + IIxI12)Mn(dx) = 0
if and only if
'Q(An) - -9'Q(µ) Proof. Let
[an, Rn, Mn] := ` (IL,,)'
[ax, R°°, M°°]
Q(µ)
[cf. (3.6.9) to (3.6.11) and (3.6.17)]. Also, for a Levy measure M and
133
RANDOM INTEGRAL REPRESENTATION
E > 0, let T,M,E be the operator defined by
f (Y, x)2M(dx) for y E Rd,
(TM.EY, Y)
(3.6.20)
where BE:_(xERd: 11x11<e). Sufficiency
From Corollary 1.8.16, we need to show that the following conditions:
(a) limn_ Jf(x)M,,(dx) = Jf(x)M(dx) for every f which is continuous, bounded, and vanishes in some neighborhood of zero, and
lim sup f log(1 + IIxII2)Mn(dx) = 0; SHOD n21 B`
(b)
lim limsup((Rny, y) + (TM",EY, y)) CIO
n- w
= lim liminf((Rny, y) + (TM",Ey, Y))
CIO n--
=(Ry,y) for all yE08d; 11man=a
(c)
n-w imply that
(a') limn _m Jf(x)M,-(dx) = Jf(x)M°°(dx) for every f which is continuous, bounded, and vanishes in some neighborhood of zero;
(b')
lim limsup ((Rny, y) + CIO n-w
= lim
CIO nom(R°°Y, Y)
(c')
y>)
y) + (T Ey, y)) "
for ally E Rd;
lim an = a',
n-+m
where Mn, M°°, Rn, an, and a°° are given by (3.6.9) to (3.6.11).
OPERATOR-SELFDECOMPOSABLE MEASURES
134
Let f be as given in (a'). Then
f f(e-`Qx) dt,
g(x)
x E lid
0
is continuous because f(e-`Qx) is zero for all sufficiently large t so, for each x, g(x) is obtained by integrating a bounded continuous function of (t, x) over a bounded 1-interval. Since f vanishes on some ball Bs about zero and, for some a > 0, Ile-'Q11 0, we see that g must also vanish on the ball BE/a about zero. Also, there is K > 0 such that, for all x,
If( x )) 5 1 K11x112 +IIzII2 because f is bounded and vanishes around zero. Thus, by Lemma 3.4.7 for all x, Ig(x)I 5 Kc6log(1 + IIz112). Therefore, using Lemma 3.6.12, lim
n-oo
f f(x)Mn(dx) = lim f f f(e-`Qx) dtMM(dx) rz w 0
gx f Mn(dx) n- oz log(, + I l x l l)
= lim =
f log(1g(x) + 114
2
M(dx) = f f(x)M°(dx). 2)
Thus, (a') holds. Let Sn,E
Rn + TM.,-1 00
Sn
E = f e-`QSn.Ee-`Q* dt, 0
IJ,E(Y) := (f0 f (IB(e-`Qx) - IB,(x))(e-`Q*y, x)2Mn(dx) dt
RANDOM INTEGRAL REPRESENTATION
135
f o r y E R ' Since ..
JS(x)M°°(dx)
= JJg(e-`Qx)M(dx) dt, 0
from (3.6.20) we see that
(T,"-,Y, Y) = JW JIB.(e-`Qx)(e-Q*y, x)2M(dx) dt. 0
Hence,
Y, y) = J(e tQSn
(IZny, y) + (T
,,e-rQ*Y,
Y) dt + In,e(Y)
0
= (Sn,EY, Y) + In,E(Y)
We will show In, E(y) -f 0 as n
--> 00
(3.6.21)
and then s J.O. We have
IIn,,(y)I < I,,,E(y) + In E(y), where In,E(y) :=
JJIB,(e-`Qx)IBE(x)(e-Q*y, x)2Mn(dx) dt, 0
(IB,(e-'Q x)IB(x)(e-'Q*y, x)2Mn(dx) dt.
In E(y) Jo
Let
k(x, t) :=
1
lie-`Qx1I2 Ile_rQxII2
+
for x c- Odd and t E R+.
Then (y) < (1 + e2) IIYII2 f J1Bes
Clearly, one may replace the norm II I{ by the norm II IIQ given by (3.4.3) (cf. Proposition 3.4.3). Let
fn(t) := Mn({x: IIXIIQ > t})
for t > 0. Hence,
s
xII QM,,(dx) = fn(s)log s + f
sS
dt.
141
RANDOM INTEGRAL REPRESENTATION
Since, for s > 1,
2 fr,fn(t)t-' dt >_ 21
fn(t)t-' dt >_ fn(s)log s,
we see that
lim sup f
1ogII xII QMM(clr) = 0
n-'0°nz1 IIxIIQZs
iff lim sup f fn(t)t-' dt = 0. s`'°° nzl $
(3.6.28)
Note that from (3.6.10) and the properties of polar coordinates m Mn([SQ' [t,-)]) =a f:1l0 M,,([SQ,
[r°`,co)])r-' dr
and
fn(r) < M,,([ SQ, [ r a, 00)j )
for r > 1 and a := IIQIIQ' (cf. the last part of the proof of Theorem 3.4.5). Hence,
lim sup f fn(t)t-' dt s lim sup f Mn([SQ, [t",c)j)t-' dt s-'0 nil s = a-' lim supMn([SQ, [s0,0.)}) = 0, s-a oo nz1
s-'oo nil s
because of (3.6.24). This together with (3.6.28) gives (3.6.27). Finally, for any strictly increasing sequences {n'} of positive integers, there is a
subsequence (n") and [a', R', M'] in ID(l ') such that
[ae, Rn"+ Mn'I - [a', R', M'] [cf. (3.6.26)], and also by (3.6.27)
lim sup f
log(1 + IIx142)Mn..(dx) = 0.
$-*00 n"2:1 IIxII>s
OPERATOR-SELFDECOMPOSABLE MEASURES
142
Thus, the first part of this theorem gives 7Q([a., R,r,
.9
([a', R', Mil) = .9 ([a, R, M])
and Proposition 3.6.10 implies that [a', R', M'] = [a, R, M]. ConseQ.E.D. [a, R, M] (cf. Corollary 1.6.3). quently, [a,,, R, Remark 3.6.14. In the preceding section, we found the generators for the class L0(Q). From Proposition 1.8.10, we have that the class ID(R') is generated by the Gaussian and Poisson measures. Since L0(Q) c ID, it is natural to consider how these generators are transformed by the mapping JQ. First, consider a E Rd \ (0) and look at
6'. There are unique z E SQ and s > 0 such that 4(z, s) = a. From (3.6.10), for r > 0 and A E q(SQ),
8-([ A, 0o)]) fo 34(I,S)([A, [etr,oo)]) dt = Sz(A)f{t>0:s>e'r} dt = SZ(A)log(max(1,
= 3,(A) f MI(o,Sj(t)t-' dt. r
Hence, for A > 0 and [r, u) c (0, oo), u
[r, u)]) = ASI(A)f I(o,s](t)t
dt,
so, for arbitrary F E R(Rd \ (0)), fs fo
IF(tQx)t-1 dt ASZ(dx)
Q
[cf. (3.5.5) and Theorem 3.5.5]. Second, for a covariance operator R, we have R°° = f e-tQRe-tQ* dt by (3.6.9), so R = QR°° + R°°Q* by Theorem 3.4.2. Therefore, these
facts raise the possibility that all of L0(Q) is generated by these simple Poisson measures [x, 0, (AA > 0, x e Rd, and by Gaussian measures [x, R, 01 such that 0 _< QR + RQ* (cf. Theorem 3.5.5).
Proposition 3.6.10 and Corollary 3.6.11 give algebraic properties of L0(Q) with respect to some linear operators, either commuting with Q
RANDOM INTEGRAL REPRESENTATION
143
or invertible. From the random integral representation of measures in L0(Q), although it is possible to use the definition of L0(Q) directly, we will see how this class is unchanged under some idempotents. Note
that in Sections 3.1 to 3.4, we worked under the very essential assumption that µ in Lo(Q) is full on (,B d. We now introduce the following notation. For a subspace W of 11 d, let
(µ E .(d): W = lin(supp µ), and, for all t
LQ(Q; W)
0,
µ =e-`Qµ * y, for some v, E 150 (Rd) with supp v, c W}. Theorem 3.6.15. If P E End(Rd) is an idempotent, that is, P2 = P, and the space ker P is Q-invariant, then for any µ E L0(Q; fled) we have Pµ E L0(PQIW; W), where W P(Rd) and PQIW is the restric-
tion of PQ to W.
Proof. Note that P lker P = 0 and ker P = (I - PXO d). Since ker P is Q-invariant, we obtain Pe-`Q(I - P) = 0 for every t e IIB', and PQ(1 - P) = 0. Hence, PQ = PQP and by induction PQkP = (pQ)kp since
PQk+Ip = (pQ)(Qkp) = (pQp)(Qkp) = (pQ)(pQkp) =
(pQ)(pQ)kp = (pQ)k+I p Consequently, Pe-`QP = e-`PQP, and e-`PQ and Pe-`Q coincide on W := P(91d). Since µ E LO (Q; Rd), we have µ = -( Joe-`Q dY(t)) for some D(tlr, [0, -))-valued r.v. Y with stationary independent increments, -Z(Y(1)) E ID,og(O '1) and Y(0) _ 0 a.s. [cf. (3.6.18)]. Hence,
Pµ =
f °°Pe-`Q d(PY(t)) +
f00Pe-`Q
d((I - P)Y(t)))
f0 e-`PQ d(PY(t))) E L0(PQIW; W),
because Pe-`Q = e-`PQ on W, Pe-`Q(I - P) = 0, and Y(PY(1)) E ID,og(W ). Since Il a-`PQII w -< sup{II e-`PQPxII: x E Rd with llxll oo),
Clearly, X' is a linear subspace of C0(Rd). From the inequality
Ilf - ARAf II s Ilf -
Ilfn - ARA fnll + IIARA(fn
- f )II
< 211f - fnll + Ilfn - ARAfnII,
we see that -q"°° is a closed subspace of C0(l '). Furthermore, if f = Rsg for some s > 0 and g E C0(Rd), then from the resolvent equation we have
[If - AR F11 = IA - si- 'IIAR A
so f c-
A
g - SR all
_ 0) forms Proposition 3.7.3. We have C0(lBd) a one parameter strongly continuous contraction semigroup (cf. Section 1.4).
Proof. Note that, for f = RAg with g E C0(I ."), we have
U,f(x) -f(x)I _
(eAt - Of e-a.Urg(x) dr (ear - 1)IIgII A
+
- f'e-x.Urg(x) dr
IIgIi(1 - e-") A
Hence, RA(Co( )) c TO so Rc eZ-0. By Propositions 3.7.1 and Q.E.D. 3.7.2, CO(Rd) = RC _q"0.
With the same proofs all of this can be generalized as follows. Remark 3.7.4. For an arbitrary locally compact space K with a countable base, each one-parameter contraction semigroup (U,: t >- 0) on CO(K), the space of all continuous functions on K which vanish at infinity oo, the one-point compactification, satisfying
lim U,f(x) = f(x) for each f E CO(K) and each x E K, also satisfies
lim U, f = f in CO(K) for each f E CO(K ), that is, it is a strongly continuous one-parameter semigroup. With the semigroup (U,, t >- 0) on CO(l ') [cf. (i), (ii), and (iii)], we
associate the operator 1' defined on the dense subset M of CO(Rd)
OPERATOR-SELFDECOMPOSABLE MEASURES
148
t-'(U1 - I) for
[cf. Propositions 3.7.1(f) and 3.7.21. Denoting by rr
t > 0, we define two other operators F0 and ro0 on the sets .
and
Roo, respectively. Namely, { f E Co(R'): there is a g f E CO(0.") such that
Ro
I't f(x) - gf(x)I = 0 for each x r= Rd and supllrrf II < c}, r>0
R.
{ f E Co(Q8d): there is an h f e C0(Rd) such that
lim Ilrf - hf11= 0},
r-.o
rof == gf and roof
hf.
The relations among these operators are given in the following proposition.
Proposition 3.7.5.
We have .1' = .JPO = Roo and IF = r0 = roo, so
rf = limr_0t-'(Urf - f) in C0(Rd)
Proof. Obviously, Roo c R. and roo = r0 on .9Poo. Now, let f E R, so f = Rag for some g E C0(Rd) and A > 0. Since, by Proposition 3.7.1(f),
rf=Af-RA'f=ARAg-g, we obtain
F,f(x) - rf(x) 7
= t-' (eAt - 1 - At) f e-ArU,g(x) dr foe-ArUg(x) drj
+ f r(g(x) - e-ArUrg(x)) dr - At =t
1((eAr ll
_ 1 - At) fe-ArUrg(x) dr + fre-ar(g(x) - Urg(x)) dr r 0
+g(x) f r(1 - e-`') dr - At f re-A'Urg(x) drj. 0
0
Hence,
Ilrrf - rfII s
(ear - 1 - At)IlgII A
At
+ Ilgllt-'
r
+ t-' fe-A'IIg - UrgII dr 0
fo (1
- e-Ar) dr + AIIgII fre-Ar dr. 0
149
INFINITESIMAL GENERATORS
By Proposition 3.7.3, we see that the second term goes to zero as t -* 0. Therefore, F, f -* F f in C0(l ) as t - 0, that is, .R ' c Moo and
r=r00 on M. Assume now that f .9Po. Then ro f (=- Co(U8d) and by the bounded convergence theorem we have
R,(I - ro)f(x) = RIf(x) - iimR,r,f(x) = R If(x)
limt -
two
( er
- 1 ) ftee- rUrf( x ) dr -e` fore-rUrf(x) drI
= f(x)
Thus, f = R1(I - ro)f E RI(C0(!Bd)) = .9P, that is, W0 c .W and ro = Q.E.D. I - RI' = r on Remark 3.7.6. Proposition 3.7.5 holds with the same proof on Co(K), where K is locally compact with a countable base (cf. Remark 3.7.4).
The operator r, or F0 or 1700 in the case CO(K), is referred to as the infinitesimal generator of the one-parameter semigroup (U,: t >: 0).
It must be emphasized that r with its domain M completely characterizes the semigroup {U,: t z 0) (cf. Corollary 1.4.3). 3.7.2
Infinitesimal Generators of Ornstein-Uhlenbeck
Type Processes
Let Y be a D(Rd' [0, oo))-valued random variable with stationary independent increments and Y(0) = 0 a.s. For t > 0 and x E 1W', let ZX(t)
e-'Qx +
le-SQ dY(s)
(3.7.3)
40,
We refer to Zs as an Ornstein-Uhlenbeck type process since, when Y
is Brownian motion, Z, is called the Ornstein-Uhlenbeck process. Our assumption on Q requires a-`Q - 0 as t - oo, so we see that a measure µ is in LO(Q) if and only if it is the limit in distribution as t - oo of some Ornstein-Uhlenbeck type process. Moreover, this limit exists if and only if ../(Y(1)) F ID,og (cf. Theorem 3.6.6). The stochastic properties of the ZX process are given in the following proposition.
OPERATOR-SELFDECOMPOSABLE MEASURES
150
Proposition 3.7.7. The Ornstein-Uhlenbeck type processes are stationary Markov processes with independent increments and with paths in D(Rd, [0, 00)).
Proof. From Lemma 3.6.3, we see that Z, has independent incre-
ments and paths in D(Rd, [0, oo)). For t z 0, x E OBd, and A E g(OBd), let
-/'(Zz(t))(A)
P(t, x, A)
From (3.7.3), we have P(0, x, A) = 5x(A). Clearly, P(t, x, - ) is a probability on q(Rd) and'P(t, - , A) is a Borel-measurable function.
To prove the proposition, we show that, for t >_ 0, s >_ 0, x E Rd, and
A E g(Rd), P(t + s, x, A) = f P(t, y, A)P(s, x, dy).
(3.7.4)
For the moment let v denote the measure given by the right side of (3.7.4). From (3.7.3) and Lemma 3.6.4, we have
P(t, x, -)(z) = exp{ i(e-`Q*z, x) + f Clog .i(Y(1))(e-`Q*z) dr}. Consequently, for z E OBd,
v(z) = f f e`(2.v)P(t, y, dv)P(s, x, dy) = f P(t, y, -)(z)P(s, x, dy) l'ei<e`Q*z. v>
J
exp (rlog
x, )(e-`Q*z)exp{ f Clog -;-(Y(1))(e-'Q*z) dr) =
exp(i(e-(s+,)Q*z, x) 11
+
f0 '+'
l
log 2(Y(1))(e-'Q*z) dr) 1
= P(t + s, x, -)(z), which establishes (3.7.4).
Q.E.D.
INFINITESIMAL GENERATORS
151
Before we begin the search for the infinitesimal generator associ-
ated with Z, note that as x --> x and t, --+ t > 0 with t >_ t, we have Zx
Zx(t). Therefore, under these conditions
P(t,,, x,,, -) - P(t, x, ).
(3.7.5)
For any real-valued bounded Borel-measurable function f on IIBd, let
Ef(Zx(t)) = f f(Y)P(t, x, dY)
(U,f)(x)
= f f(e-`Qx + y)P(t, 0, dy)
(3.7.6)
for t >_ 0 and x E R . From (3.7.4), we have that U (U,: t >_ 0) is a one-parameter semigroup of operators, that is, U,(UJ) = U f, and P(O, x, A) = 8x(A) implies that Uo f = f, and U0 = I, the identity operator. Furthermore, if f E CO(D '), that is, f is continuous and vanishes at infinity, from the last inequality in (3.7.6) and the Lebesgue bounded convergence theorem, we see that U, f E Co(Rd); thus, U,: C0(Rd) - C0(Rd),
t > 0,
are bounded linear operators on the Banach space C0(12; d) with the supremum norm. Clearly, if f >_ 0, then U, f >_ 0. Since P(t, x, ) are tight measures, their supports are cr-compact, so one can find an increasing sequence {Kn) of compact sets such that P(t, x, 1. Selecting an f,, E C0(Wd) such that 0 < f,, < 1 and I for all y E K,,, we see that IIU,II >_ f
x, dy) ? P(t, x, K,a)
Clearly, U, < 1, so this implies that U, = 1 for all t >_ 0. Since Zx is t - -.,-'(Zx(t)) are D(-60(R d),
D(IIBd, [0, oo))-valued, the functions
[0, oo))-valued, and, in particular, for any real-valued bounded continuous f on IfBd
lim(U, f)(x) = f(x) for each x E Rd.
(3.7.7)
In fact, the functions t -, (U, f Xx) are MR, [0, -))-valued. For f E
152
OPERATOR-SELFDECOMPOSABLE MEASURES
Co(l), Proposition 3.7.3 gives that (3.7.7) is equivalent to lim Ut f = f in C0(Pd).
(3.7.8)
t-0
Summarizing the above, we have U = (U,: t >_ 0), with U, given by (3.7.6), is a strongly continuous one-parameter semigroup of operators
on C0(fl8d) with IIUtII = 1 and 0 < f s 1 implies 0 < Ut f < 1. Such semigroups U are called Feller-Dynkin semigroups. Our aim now is to find the infinitesimal generator of U, that is, an operator F and its dense domain _9(I') in Co(l) such that, for f e _9(0,
Ff
lim t-'(U, f - f) in Co(Rd).
(3.7.9)
In view of Proposition 3.7.5, we see that this is equivalent to the following conditions:
supr-'IIUtf - fII < 00 and liimlt-'[Uf(x) -f(x)] t>0
for x E l
- Ff(x)j = 0 .
(3.7.10)
Now, we formulate our main result of this section. Let CK(Rd) denote the space of all twice continuously differentiable real-valued functions on Rd with compact support. Let Df(x) and D2f(x) denote
the first and second derivatives, respectively, of f at x, and tr A denote the trace of the linear operator A on Rd. Theorem 3.7.8. F o r x E 1 d, let {ZX(t): t >: 0) be the process given by the random integral (3.7.3), where Y is a D(R d, [0, oo))-valued random
variable with stationary independent increments, Y(0) = 0 a.s. and _AY(1)) = [a, R, M]. Let U = (U,: t 0) be the one parameter semigroup on C0(Pd) defined by (U f)(x) Ef(ZX(t)). Then the infinitesimal generator IF of U is the smallest closed extension of the operator G on C2(9 ) given by
(Gf)(x)
(a - Qx, Df(x)) + Z trRD2f(x) +y) -f(x) - (Y, Df(x))(1 + I1Y112)-1,M(dy).
153
INFINITESIMAL GENERATORS
Proof. First, note that applying the Taylor formula to f E
CK(W'),
we have IIYII2)-I
.f(x +Y) -f(x) - (Y,Df(x))(1 + IIY112)
II oo for 1 < k, 1 s d).
159
INFINITESIMAL GENERATORS
Furthermore, let G1 be defined on F1(ll ') by the same formula as G
in the statement of the theorem. Since (3.7.11) also holds for f E F1(IIBd), we see that GI f E CORI )
Step 2. G is an extension of G 1.
It suffices to show that given f e FI(l."), one can find a sequence { fn) in CK(fd) such that II f,, - f II --> 0 and II Gf,, - G1 f II -- 0 in the norm of C0(Rd). Select h E CZ(Old) such that 0 < h < 1, h(x) = 1 for we llxll s 1, and h(x) = 0 for IIxiI >- 2. Setting f(x) have f,, E CK(Rd) and II f - f II < supu=u,,, Ilf(x)ll -> 0 as n - 00.
Furthermore, from (3.7.11), we have completes Step 2.
G1 f II = 0, which
Step 3. If f lI xll µ(dx) < oo, equivalently f
r=G.
11
1j, l Ilxll
M(dx) < oo, then
First, we prove that Ut: F1(IW') -> FI(Pd). Let M` be the Levy
measure of _f(ZX(t)). From (3.7.3), Lemma 3.6.4, and (3.6.7), we have
M'(A) = f IM(eSQA) d s
for A E g(ll d \ (0)).
0
Setting ct
supo s S 5 t II a -SQ II, we obtain
f IlyiIM'(dy) = fo` f IB;(e-SQz)llzIIM(dz)ds < ct f t f I(e:B,r(z)IIZIIM(dz) ds. 0
< ctt f
IIzllM(dz) < oo,
because of the additional assumption on M. Hence (cf. Proposition 1.8.13), we have
EIIZX(t)II = f Ile-'Qx + yllP(t,0,dy) < oo. If f E F1(R d), from (3.7.14) and the equality
(Ut f)(x) = Ef(ZX(t)) = f f(e-`Qx + y)P(t, 0, dy),
(3.7.14)
OPERATOR-SELFDECOMPOSABLE MEASURES
160
we see the processes of differentiation and integration may be interchanged. Consequently, from the Lebesgue bounded convergence theorem, we obtain akUf and ak 1Ut f belong to Co(f1d). Furthermore,
IIxIIIa;Utf(x)I d
5 He-tQII IIxII E f I akf(e-tQx + y) I P(t, O, dy) k=1
d oo is the unique measure for which U, is µ-stationary.
Proof. The limit measure
rem 3.6.6.). Letting P(t, x, )
exists since .I'(Y(l)) E ID,,, (cf. Theo_AZx(t)), we have, for f e Cb(ll ),
J f (y)P(t + s, x, dy) = fRef Ref (z)P(t, Y, dz)P(s, x, dy) (cf. Proposition 3.7.7). Letting s -> oo, we obtain
fRef(Y)p(dy) = J
(Uf)(y),.
(dY),
which shows that U1 is µ-stationary. Suppose there is another measure
v such that U, is v-stationary. Then, for all t > 0 and for all f e= C0(l d),
fRdf(Y)v(dY) =
fRefRef(z)P(t, Y, dz)v(dY)
Letting t -> co, we obtain
JRef(Y)" (dy) = fReI f(z)lI(dz)v(dy) = fRef(z)µ(dz) for all f c- C0(lBd). Therefore, v = A.
Q.E.D.
162
OPERATOR-SELFDECOMPOSABLE MEASURES
3.8 THE ABSOLUTE CONTINUITY OF exp(-IQ) -DECOMPOSABLE MEASURES
For two measures p p21 we say that p, is absolutely continuous with respect to p21 p, _ 1, let µk be the compound Poisson measure on R d given
by µk = e(MI Bk). Letting
M(Bk) and vk
Ck
ck IMI Bk, we
obtain W
Nk = e-`k
j=0
(j!)
Ckvk
Set
akI
a":
ak2
a
n-I
(j!)-lckvk(Bk),
j=0 00
ck F, (j!) ICkvk(Bk), j=n
(3.8.5)
164
OPERATOR-SELFDECOMPOSABLE MEASURES
where n is such that M' As, x) is a real analytic function on R'. So, if 11(Zx) > 0, we x) = 0. Therefore, have
D(xEE Rd1:l1(Zx)>0) _ {X E Rd-1: f(s, x) = 0 for each s E
fE81}.
Since f is not identically equal to zero, there is s E 181 such that ZS :_ (x E 1$d-1: f(S, x) = 0) * pd-1 Note that Zs is the set of zeros of a real analytic function on Rd-' which is not identically equal to zero. Furthermore, since we have
ld(Z) =
fw-,ll(Zx)ld-1(tX)
f ol1(Zx)ld-1(dx) < f-ll(Zx)ld-1(dX), Z,
it suffices to show that ld-1(2) = 0. Since the lemma is true for d = 1, by induction, we see that it holds for any d >_ 1.
Q.E.D.
Lemma 3.8.5. Let Q be an invertible linear operator on 18d and let V be a k-dimensional subspace of 18 d with k < d such that lin Q(V ), the
smallest Q-invariant subspace of Pd containing V, coincides with Rd.
Then there are r z 2 positive integers k = j1 >_ j2
jr and
subspaces V1, ... , Vr with V1 = V such that
(1) 18d=V1®...®Vr,dim Vm=j,, for t<m 0 for some proper Q-invariant subspace V of ld. In the case of (i), µ is a full Gaussian measure on bid, so µ 0 and G(W) = 0 for every proper subspace W of W (cf. Remark 3.8.7). Let µ, :_ .7Q[0, 0, GI W ], where GI W is the restriction of G to W.
Then µ, E L0(Q). Clearly, supp(GIW) c W. Suppose linQ W is a proper subspace of Or'. Then G(linQ W) = 0 by (ii), so G(W) = 0, a
OPERATOR-SELFDECOMPOSABLE MEASURES
174
contradiction. Hence, lin Q W = fd. By our selection of W, (G I W )(W ) = G(W) = 0 for every proper subspace W of W. Thus, by Proposition µ, * .TQ[b, T, GIW`], we have it 0. Since QV = V, we have e`QV = V for all t >- 0, so by (3.8.12), M(V) = co. Let V1 := lin (supp(G I V )) and Al = .TQ[0, 0, GIV1]. Note that V D V1 D supp(GIV), so that GI V, =
GIV. We have that V1 is a Q-invariant subspace of V, dim V, < d, Al E L0(QIV,), and the Levy measure of µ, is MIV, by Remark 3.8.8.
Then µ, = [c, 0, MIV1] for some c in V, by (3.6.11). From Proposition 3.4.10,
lin(supp µ,) = lin(supp(µ1 * S(-c))) = lin(supp(MIV1)) = linQ(supp(GIV1)) = V1.
Hence, µ, is full on V,. Considering µ, as in .9(V1), we have that µ1 is full and exp(-t(QIV1))-decomposable on the r-dimensional space V1 with r < d. So, by the induction hypothesis, we see that µ, 0. Thus, the Laplace transform of the function t JRdU,f g1d is equal to the constant a, so, for all f E -9(F), fRdf g1d = fRdv,f gld.
Since this equality extends to Co(l8d), we see that the semigroup Uf, t >- 0, is v-stationary, where v(dx) := g(x)ld(dx). By the uniqueness of Proposition 3.7.10, we see that v is exp(-tQ)-decomposable with g for a density. Finally, to see that G* is of the form (3.8.13), it suffices to check that 4 dGh, - h2 dx =
hl - G*h2 dx fRd
177
MULTIVARIATE SELFDECOMPOSABLE MEASURES
for hl, h2 E CK(l ). The above is easy to see when both G and G* Q.E.D. are expressed by coordinate partial derivatives. 3.9 MULTIVARIATE SELFDECOMPOSABLE MEASURES
A probability measure µ (on a Banach space) is said to be multivariate selfdecomposable, or simply selfdecomposable, if there exist a sequence of probability measures on {an) of positive numbers, a sequence the Banach sphere in question, and a sequence (v,,) of vectors in the same Banach space such that the triangular array of measures {7
v1: 1 < j < n) is uniformly infinitesimal
(3.9.1)
and
Ta^(VI* ... *vn)*5(V.)
µ.
(3.9.2)
Later, Lo denotes the class of all selfdecomposable measures and it
is often called the Levy class of probability measures (not to be confused with Levy spectral measures of infinitely divisible measures; cf. Section 1.8). A quick look at Section 3.1 leads us to observe that selfdecomposable measures are operator-selfdecomposable ones with normings by An := a I, an > 0. Thus, all characterizations and properties discussed so far can be specialized to this class, but often only for full measures. In fact, fullness can be replaced by nondegeneracy,
because of the very particular form of An's and the convergence of types theorems discussed in Section 2.7. In particular, in case of nondegenerate limits, Corollary 2.7.3 is used in the same fashion as Theorem 2.2.10 was utilized for operator-selfdecomposability in Section 3.2. It might be worthwhile here to call attention to Remark 2.7.5. It implies that allowing an < 0 in (3.9.2) would lead to limit measures which are shifts of symmetric measures; thus, restricting the class Lo.
Proposition 3.9.1. If µ is nondegenerate and selfdecomposable, then in (3.9.2) we have an -+ 0 and 1 as n -b oo.
Proof. Proceed along the lines of the proofs of Proposition 3.2.2 and (3.2.2) using convergence of types theorems from Section 2.7. Of course, some of the steps in this case are trivial. Q.E.D.
OPERATOR-SELFDECOMPOSABLE MEASURES
178
Theorem 3.9.2. A measure µ on X is selfdecomposable, that is, µ E Lo if and only if, for each 0 < c < 1, there exists µc E .6' such
that µ = TCµ * µc.
(3.9.3)
Proof. Since degenerate measures are selfdecomposable and satisfy (3.9.3), we assume that µ is nondegenerate. From Proposition 3.9.1, for each 0 < c < 1, there exists a sequence k >- n such that ak /an Ta (v1 * .. * v,,) * 5(vn), we have that --> c as n --> oo. Setting p,,
pn - µ and Pk. = Takn/a.Pn *[i' (v
1*
... *
s(wn)]
(3.9.4)
for some wn E X. With Theorem 1.7.1(a), this implies that the second factor in (3.9.4) is conditionally compact. Denoting its limit point by µc, we get the decomposition µ = Tcµ * µ, which proves the necessity of (3.9.3). [In fact, the second factor in (3.9.3) converges to µJ For the sufficiency of (3.9.3), let us set v1 µ, vk = N-(k-1)lk, k >- 2, using (3.9.3) with c = (k - 1)/k. Taking an n -1, we obtain Tan(v1 * v2 * ... * vn) = µ.
[Note that vk = 4/(T(k_1)/kµ) ' for k > 2 and use Fourier transforms to show the above equation.] Q.E.D. The factorization, or decomposition, in (3.9.3) is the justification for the term "selfdecomposable" measure. Also, writing it in the form
µ=Te-,µ*v, for allt>0,
(3.9.5)
we see that Theorem 3.9.2 is analogous to the fundamental characterization of operator-selfdecomposable measures given in Theorem 3.5.5, due to Urbanik. It is very interesting how much deeper analysis was needed to establish Theorem 3.5.5, while Theorem 3.9.2 is a straightforward calculation.
Theorem 3.9.3 (Random Integral Representation). A measure µ is selfdecomposable if and only if there exists a D(X, [0, -))-valued random variable Y such that Y has stationary independent increments, Y(0) = 0
179
MULTIVARIATE SELFDECOMPOSABLE MEASURES
a.s., E log(1 + IIY(1)II) < oo, and
i(f0 _e -S dY(s)) = µ.
Proof. The proof goes by the same steps as the proof of Theorem Q.E.D. 3.6.8. Corollary 3.9.4. A measure µ on a Banach space X is selfdecomposable if and only if, for x* E X*,
µ(x*) = exp(i(x*,a) - 4-1(x*,Rx*) eiw<x*, x> - 1 ((
+fx"{o}((o,»
K,
l1
dw - i(x*, x)IB(x) I M(dx), (3.9.6)
where the triple a E X, R is a Gaussian covariance operator, and M is a Levy spectral measure satisfying the condition fjixjj >I log(1 + IIxIDM(dx) < oo and they are uniquely determined.
Proof. Using Lemma 3.6.4, compute µ from the random integral representation in Theorem 3.9.3, with _Z(Y(1)) = [ao, R, M]. Because of Proposition 3.6.10, _,I(Y(1)) is uniquely determined. Q.E.D.
Besides the above, other properties of selfdecomposable measures can be derived from those of operator-selfdecomposable ones. However, let us focus on the problem of distinguishing selfdecomposable measures among those which are operator-selfdecomposable. Note that (3.9.5) characterizes the class Lo as
µ =e-``µ*v, for all t> 0. In other words, I is a U-exponent of the selfdecomposable measure µ, that is, a-" E D(µ) for t > 0 (cf. the second part of Section 3.3). The rest of this section deals only with X = Rd, a finite-dimensional linear space.
Proposition 3.9.5. Let A be a measure on lid such that (0) and Rd are the only invariant subspaces with respect to the symmetry semigroup
A(µ). If p. is full and Y(D(µ)) \ .Y`(A(µ)) 0 0, then µ is selfdecom-
OPERATOR-SELFDECOMPOSABLE MEASURES
180
posable. Moreover, for a nondegenerate selfdecomposable µ, we have
-I E .T(D(µ)) \ .T(A(µ)). Proof. Since µ is full, A(µ) is a compact subgroup of Aut(08d) (cf. Corollary 2.3.2). Furthermore, by Corollary 2.4.2, there exist a subgroup ®o of the orthogonal group &= 0(Rd) and a positive-definite
symmetric W such that A(µ) = W0oW-'. Since A(W-'µ) = 00, D(W-'µ) = W-' D(A)W, and W-'µ is full, we may assume without loss of generality that A(µ) = Bo (cf. Lemma 2.3.3 and Corollary 2.1.4). Let Q E 97(D(µ)) \ .T(A(IL)) and let Q, be the commuting exponent of µ, from Proposition 3.3.8(b). For A E A(µ), A* =A-', so Q* also commutes with A(µ) as well as QCQ*. By Schur's lemma [cf. Lange (1975)], we conclude QCQ* = A21 and Q*Qc = p21 for some real A and p. Hence, A2QC = (QcQ*)QC = Qc(Q*Q,) = p2QC, so (A2 - p2)Q, = 0. Consequently, QCQ* = Q*Qc and both Qc, Q* commute with A(µ). Once again by Schur's lemma, we obtain Q, = AI. Since D(µ) is a compact subgroup of End(08d) (cf. Theorem 2.3.1) and exp(-At1) E D(µ) for t >- 0, we see that A >- 0. Furthermore, from
dA = tr(Qc) = f
tr(g-'Qg)H(dg) = tr(Q),
A(µ)
we see that A > 0, because otherwise det(e-`Q) = e-`("Q) = 1, a-`Q E D(µ), and by Proposition 2.3.5 we conclude that a-`Q E A(µ) and consequently Q ,7-(D(µ)) \ ,T(A(IL)) which contradicts our assumption. Thus, exp(-At1) E D(µ) for each t > 0, where A > 0, that is, µ is selfdecomposable. Conversely, if µ is selfdecomposable, then I is its Urbanik expo-
nent, that is, -I E 9-(D(µ)). If also -I E ,T(A(IL)), then a-`I E A(µ) and consequently Iµ(x*)I = 111(e-`x*)I - 1 as t --> oo, which Q.E.D. implies that µ is degenerate. Let us conclude this section with a discussion of measures on Rd that have "large" symmetry semigroups. In fact, we will say that µ is elliptically symmetric if its symmetry semigroup A(µ) is conjugate to
the full orthogonal group e= t9(l) so that A(µ) = W-'®W for some positive-definite and symmetric W. Proposition 3.9.6.
A measure µ on Rd is elliptically symmetric with
,T(D(µ)) \ ,T(A(IL)) 0 0 if and only if there exists a probability
MULTIVARIATE SELFDECOMPOSABLE MEASURES
d -I
181
measure p on (0, oo) such that, for y E Rd log µ(Y) = i(y, x) - c1IIWYl)2
+
c2r(d)
.2JF(' +
2J ;E
2
))
(II WYIIs/2)2i X
(3.9.7)
p(ds),
log(1 + s2)
where c, z 0, c1 + c2 > 0, r is Euler's gamma function, x E IfRd, and W is a positive-definite symmetric matrix.
Proof. Since WA(A)W-' = B, µ is full on Rd and, by Proposition 3.9.5, we see that µ is selfdecomposable. Since -1 E A(µ), we obtain that µ is a translation of a symmetric selfdecomposable measure and, by Corollary 3.9.4, we have
log µ(y) =i(y,x) - 4-'(y,Ry) i [cos t(y, x) - 1]t-' dtm(dx)
+ c2f nd
to)fo
log(1 + I I X
112)
(3.9.8)
where m is a probability measure such that c2m(dx) := log(1 + IIXI12)M(dx) is a finite measure on Pd \ (0) and c2 _> 0. Note that M integrates IIXII2 on the unit ball and log(1 + s2) < s2 for s >- 0.
Assume A(µ) = e. Uniqueness of the Gaussian and Poissonian parts in the Levy-Khintchine formula for an infinitely divisible measure implies that
ARA-' = R and
A E e.
From Schur's lemma, we obtain R = c11 for some cl >- 0. To solve the above measure equation, let us define measures Hd and p on the unit sphere Sd -' and the positive half-line as images of m under the mappings x - x/Ilxll and x -p IIxII, respectively. Then Hd is the Haar measure on the homogeneous space Sd-I = e(QRd)/e(Rd-I) and its Fourier transform is given by d
Hd(Y) =
r(2)
(-1)'
jlr(j + 2)
2j 112 II
Note that 1)1(y) = coslyl and, for d >_ 2, Hd can be expressed in
OPERATOR-SELFDECOMPOSABLE MEASURES
182
terms of Bessel functions. Finally, R" \ (0) is isomorphic to Sd-' X R+
and (u, t) ---> ut is a product of two independent random variables under m which gives
m(F) = f 0OHd(t-'F)p(dt) for all Borel sets F.
(3.9.9)
0
Putting this into the formula for µ and integrating, we obtain Proposition 3.9.6 with W = I. The general case, A(µ) = W-'®W, is reduced to the previous one by the following observations:
A(WA) = 6,
D(Wµ) = W D(µ)W`',
,9-(D(B'µ)) = W.T(D(p ))W-', which completes the necessity. The sufficiency is a consequence of the observation that (3.9.7) can be written as (3.9.8) with m given by (3.9.9), thus µ is selfdecomposable; and then referring to Proposition 3.9.5. Q.E.D.
3.10 BIBLIOGRAPHIC COMMENTS
The results in Sections 3.1 and 3.2 are due to Urbanik (1972), where Rd-valued random variables are discussed. However, the proofs of all the auxiliary lemmas in Section 3.3 which lead to the fundamental characterization of operator-selfdecomposable measures given in Theorem 3.3.5 are taken from Urbanik (1978), and they are valid in a Banach space. The notion of U-exponents and their properties discussed in the second part of Section 3.3 are from Jurek (1991). The solution of the measure inequality given in Theorems 3.4.4 and 3.4.5 is from Jurek (1982b), but the polar coordinates system and the new norm are from Jurek (1984). Theorem 3.4.4 was also proved by Wolfe (1980). Theorem 3.4.6 was first proved in Urbanik (1972), but
the proof was based on Choquet's theorem on extreme points of compact convex sets. Lemma 3.4.7 is also from Urbanik (1972). Properties of the support of Levy spectral measures and Gaussian covariance operators of operator-selfdecomposable measures follow Jurek (1989a). Generators for L0(Q) in Section 3.5 may be found in Jurek (1982b). Section 3.6 on the random integral representation is based on Jurek
BIBLIOGRAPHIC COMMENTS
183
(1982c) which was a follow up of the basic paper by Jurek and Vervaat (1983). In Wolfe (1982a) and (1982b), similar results may be found,
but with different proofs. The continuity of the random integral mapping in Theorem 3.6.13 is proved in Sato and Yamazato (1984b) for Rd. A different proof which is valid in the Banach space case is given in Jurek and Rosinski (1988). The material in Section 3.7.1 on
one-parameter semigroups of operators on Co(l ') is standard and appears in Freedman (1971) and Williams (1979). The infinitesimal generators of Ornstein-Uhlenbeck type processes associated with exp(-tQ)-decomposable measures in Theorem 3.7.8 may be found in Sato and Yamazato (1984a). Section 3.8 on the absolute continuity of exp(-tQ)-decomposable measures is from Yamazato (1983), which is a generalization of Sato (1985). Theorem 3.8.1 is due to Sato. In our presentation of Theorem 3.8.1, we use the random integral presentation and Lemma 3.8.4 from Jurek (1988b). Theorem 3.8.10 is from Sato and Yamazato (1984a). The characterization of selfdecomposable
measures on Banach spaces in Section 3.9 is due to Kumar and Schreiber (1975). The random integral representation for those mea-
sures given in Theorem 3.9.3 is due to Jurek and Vervaat (1983). Finally, elliptically symmetric operator-selfdecomposable measures are described in Jurek (1991).
CHAPTER 4
Operator-Stable Measures
Certain measures µ E ID(R") have the property that, for some B E Aut(ll ), µ` = t Bµ * 8(b(t )) for all t > 0, with b(t) E Rd. These
measures are called operator-stable and B is called an exponent, and they arise in a natural way (cf. Section 4.2). Section 4.4 deals with
showing that the fixed points of the mapping JB [cf. (3.6.17) and Proposition 3.6.10] are the operator-stable measures with exponent B. In Section 4.6, a decomposition of a full operator-stable measure into
its Gaussian component and its compound Poisson component is explicitly obtained in terms of the decomposition of Rd by "eigenspaces" of the exponent B. Also, in Section 4.6, the class of all exponents of a full operator-stable measure is described. Section 4.8 deals with the special case of symmetric measures. In Section 4.9, it is shown that all full operator-stable measures are absolutely continuous. This follows from the fact that full operator-selfdecomposable measures are absolutely continuous, but a simpler proof is presented. The question of moments is dealt with in Section 4.12. Domain of attraction problems are treated in Sections 4.13 and 4.14. Finally, some special cases are considered in Section 4.15.
4.1
STATEMENT OF THE PROBLEM
In the preceding chapter, we considered limit measures obtained from
A,) * 3(a,), where each A is a linear operator on Rd, 1 < k < n, 1 < n} is an infinitesimal system. We now specialize to the important case when An = v for all n >_ 1, that is, independent identically distributed summands. Most results require that the limit measure be full. With this assumption, we see that each A,, is invertible and v is full (cf. Corollary 2.2.1).
each a E II", and
185
OPERATOR-STABLE MEASURES
186
Also, (Anv: n z 1) is an infinitesimal system. To see this, note that the assumption Anv" * 8(an) - µ implies that An(v°)n - µo (The notation v", where v is a measure and n is a positive integer, means the n-fold convolution of v with itself.) Since (µ°)^ (0) = 1 and (µ°)^ is continuous, there is a > 0 such that b°)^(x)1 > 0 for all 11x11 - 1) is also an infinitesimal array. Thus, the class of all operator-stable measures is contained in the Rd.
class of all operator-selfdecomposable measures.
4.2
SAKOVIC-SHARPE CHARACTERIZATION
We call a measure µ E .J0 operator-stable provided there are a measure v E -60, a sequence (An) of linear operators, and a sequence
(an) of vectors such that Anv" * S(an) - µ. In terms of random vectors, is called operator-stable if there is a sequence (in) of independent identically distributed random vectors which converges in law to when suitably normed and centered. Sharpe's basic charac-
terization of such full µ is that A' = t'µ * S(b(t)) for all t s 0, for some linear operator B and vectors b(t). Before obtaining this charac-
terization for all t s 0, we show that its analog is valid for positive integers. Theorem 4.2.1. Let µ be a full measure. Then µ is operator-stable if and only if, for every n >_ 1, there are linear operators Bn on Rd and
vectors bn in Rd such that An = Bnµ * 5(bn), that is, µ and µ" are of the same type.
Proof. Assume µ" = Bnµ * S(bn) for every n > 1. Since µ is full, so are the An for all n >- 1. Hence, each Bn is invertible (cf. Corollary 2.1.3). Solving µ" = Bnµ * S(bn) for µ, we obtain µ = BR 'µ" * 5(-B,-,'bn). Thus, µ is operator-stable. Now, assume µ is operator-stable. Then there are v e ,91(l8d), Ak E End, ak e Rd such that Akvk * S(ak) - A. Since µ is full, we may assume that Ak are invertible (cf. Corollary 2.2.1). For n 1, An = (limk -.,, Akvk * 5(ak))" = llmk --.- Akvk' * 8(nak). Also, µ = Aknvkn * S(akn). limk Setting P'k := Aknvkn * S(akn), we have µk - A. But, with Ck := AkAkrt and Ck := nak - AkAknakn, we also
187
SAKOVIC -SHARPE CHARACTERIZATION
µ". Since p. and µ" are full, have Ckµk * S(Ck) = Akvk" * S(nak) they are of the same type by the convergence of types theorem 2.2.10 Q.E.D.
Remark 4.2.2
(a) Each full operator-stable µ is infinitely divisible.
(b) The B" in Theorem 4.2.1 is a limit point of the sequence {AkAk,;: k a: 1), where (Ak) is a norming sequence of the measure µ.
This follows from the preceding theorem since, for every n >- 1, Also, operatorstable measures are defined as limit measures of infinitesimal systems
µ = Bn'µ" * S(-Bn'b,,) = (B"'µ *
S(-B"'b"/n))".
so that they must be infinitely divisible. Let OS(R) denote the class of all operator-stable measures on 68d Occasionally, we simply write OS. Since the characteristic function v of an infinitely divisible measure v has the property that v"', for t > 0, is again a characteristic function, we let v` denote the corresponding infinitely divisible measure. The desired characterization for full oper-
ator-stable measures is obtained through a series of lemmas. For t > 0, let Gf := (A E Aut(Rd): for some a E Rd, µ' = Aµ * 3(a)), and let G U , > 0G1. We show by the following lemmas that G is a closed subgroup of Aut(l8d) and that G, = A(µ), the symmetry group of µ, is a compact normal subgroup of G. Lemma 4.2.3.
For all t > 0, G, * 0.
Proof. By the previous theorem, Gt
0 when t is a positive
integer. For now, assume t is the ratio of two positive integers k and j, so t = k/j. Since Gi # 0 and Gk _* 0, there are Aj. Ak, ap ak such that µi = A.µ * S(a1) and µk = Akµ * 3(ak). Solving the first equa-
tion, we have µ = A7'µ' * S( -Af 'a), and, substituting into the second equation, we obtain µk = AkA- 'µ' * S(ak - AkAT 'a;), that is, µ' and µk are of the same type. Let B Ak A7 ' and b ak AkA, 'ap so µk = Bµ' * S(b). Thus, µk = (Bµ * S(b/j))', so Biz * S(b/j) is the unique j" root of the infinitely divisible measure µk. Therefore, Bµ * S(b/j) = (µk)"J = µ`, so G, * 0 when t is a positive rational number. Now, for t a positive irrational number, select 0 < t,, - * t with each t" rational. Then µ'^ - µ'. Since each Gt * 0, there are B,, and h" such that µ`^ = B"µ * S(b"). Hence,
OPERATOR-STABLE MEASURES
188
= B,- 'Al- * S( -B 1bn). By the convergence of types theorem 2.2.10, Q.E.D. µ and At are of the same type, so G, # 0. Lemma 4.2.4.
For all s, t > 0, Gt-1 = G1,t and GS, = GSG,.
Proof. Let t > 0 be fixed. Then B E G1,,, if and only if µ111 = Bµ * S(b) for some vector b if and only if At = B-'µ * S(-tB b) if and only if B-' E G,. Therefore, G11, = Gr 1 for all t > 0. Let B E GSG,. Then B = AC for some A E GS and some C E G,. Hence, µts = (Aµ * S(a))' = AA' * S(ta) = ACA * S(Ac + ta), for
some a and c in Rd. Thus, B = AC E G,S, so GG, c G,S for all s, t > 0. To obtain the reverse inclusion, set s = 1/u and t = uv to obtain G1luGu,, c G. Then Gu, c GuG for all u, v > 0. Q.E.D.
Note that G,GS = GSGt for all t, s > 0, but A E Gt and B E GS need not commute. Lemma 4.2.5.
When s# t, GSnG,=0.
Proof. Suppose that for some s * t there is A E GS n G,. Hence (µo)S =Aµ * Aµ =A(µ°) = (µ°)t. Thus, (µ°)^S-`(y) = 1 for all y e R d. Since s * t, this implies that µo is not full. This contradiction establishes the lemma.
Q.E.D.
Lemma 4.2.6. If An - A E Aut(Rd) with An E Gt then there is a unique t > 0 such that A E Gt and t,, -> t.
Proof. First, we show that 0 < inf n Z 1 to < sups z 1 to < oo. Suppose infn,1 to = 0. For convenience, we suppose to - 0 instead of some subsequence of (tn}. Then (µ°)t^ 8(0). But, (µ°)t^ = A,(µ°) A(A0) Since A is invertible, µ° is not full. Therefore, 0 < inf,, t,. Suppose SUP,, z 1 to = oo. Again, we suppose t,, -, oo, so l It, -- 0. Let Bn = A n
1,
so Bn E G 1/,^ and B,, - A -1 (=- Aut(Il 'I). The preceding argument again leads to a contradiction, so sup,, 1 to < oo. Next, let t be any cluster point of {t,: n > 1). Then t is positive and finite. Let tnk - t, so that A,,A Aµ and At-k- µt. Since A,kµ = µt^k * S(ak) for some ak, we have ak -- (some) a. Hence, Aµ = limk _,^ A,, µ = limk At^k * S(ak) = At * S(a), so A E G,. Since G, n k GS = 0 for t 0 s, we have that (t,: n >- 1) has exactly one cluster
point. Therefore, the sequence {t,} converges to some t > 0 and A E G,. Q.E.D.
SAKOVIC-SHARPE CHARACTERIZATION
Lemma 4.2.7. Aut(Pd).
The set G
U
189 t > 0G,
is
a closed subgroup of
Proof. By Lemma 4.2.4, G is closed with respect to products and contains its inverses, so G is a subgroup of Aut(Rd). By Lemma 4.2.6, G is closed. Q.E.D.
Let l denote the mapping from G to the positive reals given by 1(A) = t for A E G,. By Lemma 4.2.5, 1 is a well-defined mapping. Lemma 4.2.8.
The mapping 1 is a continuous homeomorphism from
the group G to the group R+ and the kernel of 1, G,, is a compact normal subgroup of G.
Proof. For A E G, and B E G, AB E G,5, so l(AB) = is = 1(A)1(B). Thus, 1 is a homeomorphism. By Lemma 4.2.6, 1 is continu-
ous. The kernel of 1 is simply the symmetry group A(µ), which is a compact subgroup of G by Corollary 2.3.2. To see that G, is normal, note that, for fixed g c- G,,
gG1g-1 c G,GIG,,,, = Gi = G1. Now, define L: 5'-(G)
Q.E.D.
R by
L(A) := log(l(exp A)), where T(G) is the tangent space of the group G (cf. Section 1.5). Lemma 4.2.9. The function L defined above is a continuous linear functional on _97(G).
Proof. First, we show that for all t e R and for all A E .7-(G), L(tA) = t L(A). Fix A E .7(G) and let h: 68 -* U be given by h(t) _= L(tA) for t e R. Since sA and to commute,
h(s + t) = log l(exp(sA) exp(tA)) = log(1(exp(sA)) l(exp(tA))) = log 1(exp(sA)) + logl(exp(tA)) = h(s) + h(t).
OPERATOR-STABLE MEASURES
190
Hence, h is a continuous homeomorphism from R to R. Therefore, there is a E I1 such that
h(t)=ta fortEl. Since L(A) = h(1) = a, we have
L(tA)=h(t)=to=tL(A) for all tEl,all AE.l(G). Second, we show that for all A, B E T(G), L(A + B) = L(A) + L(B). Fix A, B E .T(G). By Lemma 1.5.5, the function
f(t)
exp-'(exp(tA) exp(tB))
is well defined for t sufficiently near zero, has values in _97(G), is analytic at zero, and f'(0) = A + B. Also,
L(f(t)) = log l(exp(tA) exp(tB)) = L(tA) + L(tB), since h is a homeomorphism. Therefore,
L(A + B) = L(f'(0)) = L(lim f(t) - lim 1
t) 1
= lim -L(f(t)) = lim -(L(tA) + L(tB)) t-+0 t
t-'0 t
1
= lim -(tL(A) + tL(B)) = L(A) + L(B). t
Therefore, L is a continuous linear functional. Lemma 4.2.10.
Q.E.D.
The kernel of L is -17(G,).
Proof. Since G1 = A(µ) is a compact subgroup of G, its tangent
space is well defined. Let A E 9-(G1). Then exp(A) E G1, so 1(exp(A)) = 1. Thus, L(A) = 0 and A e ker L. For A E ker L, L(A) = 0, so L(tA) = 0 for all t E R. Thus, exp(tA) E G1 for all t e R. Since A = t-'(exp(tA) - I), we have A E Q.E.D. Lemma 4.2.11. hood of I in G.
The group G1 does not contain an open neighbor-
SAKOVIC-SHARPE CHARACTERIZATION
191
Proof. Let 1 # t --' 1. Select An E G1 for all n >_ 1. Then there 3(a,). Since µ`^ . µ, the sequence are an E R" such that p" = (An) is conditionally compact in Aut(IRd) because µ is full (cf. Lemma
2.2.3). Let (An ) be a subsequence of (An) such that A^k - (some) Since t,k - 1, by Lemma 4.2.6, we have that A E G1. AE for each k > 1. Then Ck E G1k # G, by Lemma Set Ck Q.E.D.
4.2.4. Clearly, Ck -4 I.
Summarizing the above, we have the following diagram: G
I
exp1
-9-(G)
J,iog
L
(R, +)
where (R+, - ) is the multiplicative group, (R, + ) is the additive group,
and all the mappings exp, 1, log, and L are continuous homeomorphisms between the appropriate algebraic structures (groups or linear spaces). Moreover, ker L = .T(G1) and G, does not contain any open neighborhood of the identity. Now, we are ready to present the main characterization of the class OS(Rd). Theorem 4.2.12. If µ is full and operator stable on 18d, then there are a linear operator B and a function b: (0, oo) --> Rd such that, for all
t > 0, (4.2.1).
µ' = tBµ * 5(b(t)). Conversely, any measure satisfying (4.2.1) is operator-stable.
Proof. By Theorem 1.5.6, I E G belongs to the interior of exp(Y(G)). Thus, by Lemma 4.2.11,
exp(.l(G)) t G. Let C E .17(G) be such that exp(C) 0 G1. Since h(t) = log l(exp(tC)) is a continuous homeomorphism from l to R, there is a e LR
such that, for all t e R, log 1(exp(tC)) = ta. Since a = 0 implies l(exp(tC)) = 1 for all t E R which in turn implies exp(C) E G1, a
contradiction, we see that a * 0. Let B
tER,
log 1(exp(tB)) =
(1/a)C. Then, for all
(L)a = t. a
OPERATOR-STABLE MEASURES
192
Thus, for all t > 0, log l(tB) = log t, that is,
l(tB) = t.
Thus, tB E G, for all t > 0 and there is b(t) E 18d such that At = t Bµ * S(b(t)).
-n -Bb(n) for µ, A,, n -B, and a,, A,,)* 8(an) = n-Bµn * S(-n-Bb(n)) = µ Q.E.D. for all n. Thus, µ is operator-stable.
For the converse, let µ n >: 1. Then
Corollary 4.2.13. Let µ be full operator-stable and let B be a linear operator such that At = t BA * S(b(t)) for all t > 0. Then
tB_*O ast-*0; equivalently, the eigenvalues of B all have positive real parts.
Proof. Since A' - 3(0), as t - 0, t'µ * S(b(t)) converges as t - 0. is conditionally compact in the space of all affine transformations. Let a be a limit point of (( t B; NO)), , 0 as t Since µ is full, ((t B; b(t )) }t ,
0
goes to zero on some sequence. Then a A = 3(0). Since µ is full,
a=0, so t'- 0ast-0.
Q.E.D.
Corollary 4.2.14. Let µ be full and operator-stable on 18 O. Then I2(y)I < 1 for ally e 1184 \ (0).
Proof. Let H (y E ll ': 1µ(y)12 = 1}. As in Section 2.1, H is a closed subgroup of Rd. By Proposition 2.1.1, since µ is full, H does not contain any one-dimensional subspace of Rd. Hence, H must be either (0) or a lattice. But, since p. = t BA * S(b(t)) for all t > 0, we have I2(y)I` = I,2(tB*y)I and 1µ(t-B*y)I` =1µ(y)1. Therefore, tB*H = H
for all t > 0. By Corollary 4.2.13, H cannot be a lattice. Therefore, H = (0). Q.E.D. A linear operator B satisfying Theorem 4.2.12 is called an exponent
of the measure µ and the function is called a centering function of A. From the proof of the theorem, we see that the exponent B
193
A CLASS OF 18-STABLE MEASURES
need not be unique. Later, in Section 4.6, we characterize the class of all exponents for a given A. 4.3 A CLASS OF t B-STABLE MEASURES
For a fixed linear operator B such that t' - 0 (cf. Corollary 4.2.13),
we introduce the class ./'(B) of all te-stable measures, that is, µ E ,/(B) provided, for each t > 0, p. = t8µ * S(b(t)) for some vector b(t). We have .'(B) c L0(B), where LO(B) is given in Section 3.4, because, for each t >- 1, A = t-s[Ar * S(-b(t))] = e-(Iost)BA * t-B[ pt-1 * S(-b(t))}
or µ =
e_sBp.
* v, for all s >- 0, where v, := * S(-b(es))]. The algebraic and topological structures of the class -AB) are given e-se[µes-1
in the following proposition. Proposition 4.3.1
(a) The class I(B) is a closed subsemigroup of ID. (b) If the linear operatorA commutes with B, then AF(B) c ./'(B); if, in addition, A is invertible, then A./(B) = --.""(B). (c) For each x E Rd and c > 0, Tc. '(B) * S(x) = .1(B), where Tc y := cy for y E Rd. Proof
(a) It is easy to see that is a semigroup. Now, let {µn) be a sequence in .Y(B) and An - (some) A. Then µ is infinitely divisible, ,, - p! and t Bpn - t BA as n --b oo for all t > 0. From the equality
µ;, =tsµn*S(bn(t)) for
n >_
land t> 0,
we obtain that bn(t) -* (some) b(t) E ll' as n - oo for all t > 0 (cf. Proposition 1.7.6, and Theorem 1.7.1). Therefore, AI = tBA * S(b(t)) for all t > 0, that is, µ E --e'(B).
(b) Assume A commutes with B and let µ E -,,(B). Then (Aµ)` = A(teµ * S(b(t))) = tB(AA)* S(Ab(t)) for all t > 0, so
OPERATOR-STABLE MEASURES
194
A,u E _,,'(B). In addition, when A-' exists, we have _,'(B) c c --,'(B), so A--,-(B) = -,,'(B). Q.E.D.
(c) This follows from (b).
For a measure µ = [a, R, M] to be IB-stable imposes many relationships of B with R and with M. In this section, some of these are used to obtain a representation of the characteristic function of any t B-stable measure. We begin with the most basic of these relation-
ships. Before that, we mention that the notation t N is different from tN, where t E Q8' \ (0) and N is a measure. The latter means
tN(F) := t'N(F) = N(t-'(F)) = N((x E Rd: tx E F)), whereas the former, t - N(F), means the product of the two numbers
t and N(F). Proposition 4.3.2. Let µ = [a, R, M] and let B be a linear operator. Then µ is t '-stable if and only if, for all t > 0, tR = t BRt B*
t - M = tBM.
and
Proof. Since for all t > 0 and for all y E Rd, we have (tBp * S(b(t)))^ (y) 1
= exp i(b(t) + t8a, y) - 2 (tBRtB*Y, y) i(x, tB*Y)
+ f exp(i(x, tB*y)) - 1
+ (x, x)
jM(dr)
= [a,(t), tBRtB*, tBMI A(t), where
a,(t) := b(t) + tBa + fl
x 1 + IIxII2
x 1+
and since
µ` _ [ta, tR, t M],
IIt-BxII2
jtBM(dx),
A CLASS OF t8-STABLE MEASURES
195
so by the uniqueness of the Levy-Khintchine representation we obtain the equalities.
Q.E.D.
Proposition 43.3. Let B and R be linear operators. Then, for all
t>0,tR=tBRtB*ifandonly ifR=BR+RB*.
Proof. Assume tR = t BRt B* for all t > 0. Differentiating with respect to t, we obtain R = t -Bt BRtB* + t -'t BRt B*B*. Evaluating this
at t= 1, we have R=BR+RB*. Conversely, assume R = BR + RB*. For fixed x and y, define
g(x, y, t)
(tBRtB x, y)
for t > 0.
Then
g(x, y' t) = (tBBRtB`x, y) +
(tBRB*tB*x, y)
= r a (x, Y, t)
-
Hence, g(x, y, t) = c(x, y)t for some c(x, y) E P'. Clearly, c(x, y) _
g(x, y, 1) = (Rx, y). Hence, g(x, y, t) = (tRx, y), that (t BRt Bx, y) = (tRx, y) for all x, y E Pd and all t > 0.
is,
Q.E.D.
To solve the equation t - M = t BM for all t > 0, we use the new norm on Rd introduced in (3.4.3), that is, IIXIIB
f
wile-tBxll dt
0
Since t-B - 0 as t --> 00, SB
=
f'IISBxIIS-' ds.
0
II
IIB is well defined and gives a norm. Let {x E pd: IIXIIB = 1),
that is, SB is the unit sphere in Rd with respect to this norm. As before, define
c1:SBXR'-Bpd\{0} by
'(x, t) := tBx. As usual, we suppress the dependency of 1 on B since B is fixed. Proposition 3.4.3 states that 4 is a homeomorphism and for x fixed,
OPERATOR-STABLE MEASURES
196
the function t -' II t Bx II B, t > 0, is strictly increasing. Also, recall that
for F c SB and G c (0, oo), [F; G] :_ (tBx: x E F, t E G). Hence, for s > 0, sB[F; G] = {rBx: x E F, r e sG) = [F; sG]. Proposition 4.3.4. Let G be a measure on 9(R' \ (0)) which is finite on any set bounded away from zero. Then, for all t > 0, t G = t BG if and only if
G(A) = sBf f
IA(sax)s_2
dsm(dx)
(4.3.1)
0
for all A E .Q(Rd \ (0)), for some finite Borel measure m on SB. In particular, for F E g(SB),
m(F) := G([F;[1,oo)]) Proof. Assume t G = t BG for all t > 0. For fixed F E .,d`(S8), set f(s) := G([F; [s, oo)]) for s > 0. Then, for all t, s > 0, we have
tf(s)_(t8G)([F;[s,oo)])=G(f F,(t,oo)1) _ f(t). Thus, sf(s) = f(1) for all s > 0, so G([F;[s, oo)]) = m(F)/s, where m(F) := G([F;[1, oo)]). Clearly, m is a finite measure on
0<s 0,
(tBG)(A) = G(t-BA) = f f IA((st)B)s-2dsm(dx) SB 0
= t f f "IA(rax)r-2 drm(dx) = t G(A). Q.E.D. SB 0
197
A CLASS OF tB-STABLE MEASURES
Corollary 4.3.5. Let G be a measure on 9(Rd \ {0}) satisfying G([F;[1, oo)]) for (4.3.1) for the finite measure m given by m(F) F E 9(SB). Then
supp G = U tB(supp m) U (0) t>o
and lin(supp G) = linB(supp m). [Recall that lin(C) denotes the linear subspace generated by the set C of vectors, whereas linB(C) denotes the smallest B-invariant subspace generated by C.]
Proof. For product measures, it is easy to see that supp(vl X v2) _ (supp v1) X (supp v2). Let A be the measure on 9((0,ao)) given by A(A) := JAU-2 du. Then (4.3.1) implies that G ='(m X A), so supp G = 1(supp m X [0, oo)). The proof that lin(supp G) = linB(supp m) is similar to the proof of Corollary 3.4.10. Q.E.D.
The next result concerning M is easily obtained due to the fact that t B-stable implies exp(-sB)-decomposable. Hence, from Corollary 3.4.9, we have the following corollary. Corollary 4.3.6.
Let M be the Levy measure of a t '-stable measure.
(a) If V is a B-invariant subspace of lFd, then M(V) is either zero or infinite. In particular, M(Rd) is either zero or infinite. (b) If C is a countable subset of (0, oo), then M([SB; Cl) = 0. (c) In particular, M has no atoms.
The results of this section yield the following representation of the characteristic function. Its proof is immediate. Theorem 4.3.7. The measure µ = [a, R, M] is tB-stable if and only if its characteristic function is of the form 1
µ(Y) = exp i(a, y) - 2 (RY, y) + ,Bfo a 0 and z E Rd, then p. is t`B stable
Proof. Let µ = [a, R, M I be t B-stable. Then R and M satisfy Propositions 4.3.2 and 4.3.3. Since .9B(µ) = [a°°, R°°, Mi, from (3.6.9) to (3.6.11) we have
R°° = f oe-`B(BR +
RB*)e-tB*
dt
0
=
f
dt + f
e-IBRB*e-tB*
dt = R,
where the last equality is by the integration by parts formula and the fact that a-tB -> 0 as t -> oo; also, for F E R(Rd \ (0)), M°°(F)
-f
f f °°IF((e-st)Bx)t-2 dtm(dx) ds sB 0
o W
O
1
00
fSB (o
1FF(rBx)r-2e-' drm(dx) ds = M(F);
and, finally, a°° = B- 'a, so x = (B-' - !)a. Conversely, if µ = [b, T, N] E IDIog, then -'TB(IA) = pf * S(z) for some c > 0 and vector z is equivalent to b°° = cb + z,
cT=
f
Oe-tBTe-tB*
dt, (4.4.1)
0
c N(F) = f ON(e`BF) dt 0
for all F E Q(Rd \ (0)). By integration by parts, we obtain
(cB)T + T(cB*) = f
(-e-tB)lTe-tB*
dt
0
+f(-e- tB) (Te - B* j dt = T;
so, from Proposition 4.3.3 and Remark 4.3.8, we see that T is the
OPERATOR-STABLE MEASURES
200
covariance operator of a t`B-stable measure. The last equality in (4.4.1) expressed in terms of the B-Levy spectral function (cf. Section 3.4) becomes
cLN(A; t) = f 'OLN(A; s)s-' ds
for t > 0 and A E Q(SB). Hence, dLN C
dt
(A; t) = -t-'LN(A; t)
except for countably many values of t (cf. Proposition 3.4.1 and Theorem 3.4.4). Therefore,
LN(A; t) = -cy(A)t for some positive constant y(A). Consequently, y is a finite Borel measure on SB and
N([A,[t, s)]) = y(A) f Sr-`-'-' dr r
f
I[A.(I.s)](rBU)r-`
'-1 dr y(du).
SB 0
Hence, for F E Q(D "\ {0}) and y(A)
cy(c-'A) for A E .e(SB)
(cf. the proof of Remark 3.5.6), we have
N(F) =
fS8f 0IF(r)
u)r-`-'-'
dry(du)
cJsB f IF(5CBU)S-z ds y(du) 0
_-
f
fp 00 ScB
IAScBX)S-Z
ds y(dx),
since cSB = S,,B. By Proposition 4.3.4, this implies that Tc-,N is the Levy measure of some t`B-stable measure. Finally, Proposition 4.3.1 shows that N is the Levy measure of some tcB-stable measure. Q.E.D.
201
NORMING SEQUENCES
Corollary 4.4.3. A measure [a, R, M ] is t '-stable if and only if there is c > 0 such that, for all F E R(Rd \ {0}),
M(F) = c f M(e1BF) dt 0
and
R=cf
e-tBRe-tB*
dt.
0
Corollary 4.4.4. A measure µ E ID log is t B-stable if and only if TB(A) = µ * S(z) for some z e Rd. Consequently, we have _I(B) .TB(_,,1(B)), that is, the class .,'(B) is invariant under the mapping .TB
4.5 NORMING SEQUENCES A norming sequence {An} for an operator-stable measure µ is defined
analogously to the case of an operator-selfdecomposable measure, that is, Anvn * S(an)
A
(4.5.1)
for some measure v and some sequence {an) of vectors. Therefore, all of the results in Section 3.2 apply to norming sequences {An} of an operator-stable measure; in particular, (a) An -* 0 as n -, and (b)
there is a norming sequence {Bn} such that Bn+,Bn' - I. In this section, we establish useful bounds on IIAnII and IIAn'II. Throughout
this section, µ is a full operator-stable measure with a norming sequence {An} and µ is tB-stable.
Let k and r be integers with 0 5 r < k. Then for any norming sequence (An} of a full operator-stable measure µ, the sequences (Ank+rAR'kB)",I and {k-BAnAnk+r}nzI are conditionally compact and their limit points belong to A(µ), the symmetry group of µ, where B is an exponent of µ. Proposition 4.5.1.
Proof. Let v and an be as in (4.5.1). By Proposition 3.2.1, An - 0 as n --> -, so Ank+,v' - S(0) as n --> oo. Consequently, Ank+rvnk
* S(ank+r) - µ as n - -.
(4.5.2)
OPERATOR-STABLE MEASURES
202
Also, Anv"k * S(kan)
µk as n --> . But, µk = Vg * S(b(k)) for
some vector NO. Thus, k-BA"vnk *
as n
S(k-B-'an
- k-Bb(k)) - µ
(4.5.3)
oo. The convergences in (4.5.2) and (4.5.3), by Theorem 2.2.10,
imply that k-BAn = B"C"Ank+r with Bn -- I as n -3- oo and each Cn E A(µ). Thus, Ank+rA,'ke = (B,Cn)-'. Since A(µ) is a compact group (cf. Corollary 2.3.2), we have the proposition for {Ank+rAn'k B) The remainder of the proof is obvious. Q.E.D. Proposition 4.5.2. Let k be a positive integer. Then for any norming sequence (An) of the full operator-stable measure µ with an exponent B, we have
- All: A E A(µ)} = 0
lim
max inf{II
lim
max inf{IIk-BAnAnk+r - All: A (=- A(µ)} = 0.
n-- 05r 0 such that, for m >_ m0, max lI Amk+rA,n'kRII < c1 + 1.
Osr_ m0 and 0< r < k,
IIAmk+rAn'll m0. Since mj >_ k'-', for .
? n0, IIAm;,1A;n,'II : N and for all m > 0,
IIA"A-'Il < c'bmb.
Proof. Since all norms on Rd are equivalent, in this proof
II
ll
denotes the norm II II,, defined in Section 4.7. Hence, for A E A(µ), IIAII = 1. Select q so large that 2llg8-b"ll < 1, let c := [log, m], and
write A(n) for A. Then (ClA(nqi)A(nqJ±1)_1)A(nqc)A(nrn)_1. -A"A-'m =
i=o
For 0 < j < c - 1, in Proposition 4.5.2, replace n, k, r by nq, q, 0, respectively, to obtain that the limit points of A(nq')A(nq'+')-' are in q BA(µ). Hence, for n sufficiently large and for all j = 0, ... , c -
1,
A(nq')A(nq'+1)-111
< 211gBJI.
II
Similarly, the limit points of A(nq`)A(nm)-' are in A(µ), so for n sufficiently large A(nq`)A(nm)-1Il
II
< 2.
205
CHARACTERIZATIONS OF OPERATOR-STABLE MEASURES
Hence, for n sufficiently large, 2`+'IigBll` 0) for some y E R O. For E E R(R' ), let p1(E) :_ p(E f A)p(Rd)/p(A) and P2(E) := p(E n (18d \A))p(Pd)/P(R d \A). Since the supports of p1 and P2 are 0-invariant, p, and P2 are in A2'
For a := p(A)/p( '), we have 0 < a < 1 and p =apt + (1 - a)p2, which contradicts p being an extreme point of A2. Therefore, the set of extreme points of A2 is contained in A3. The reverse inclusion is obvious.
Hence, the set of all convex combinations of measures in A3 is dense in A 1. Since there is a one-to-one correspondence between A 1
and A, linear combinations of those M E A whose supports are exactly one orbit of t B are dense in A. Finally, since t N = t BN for all t > 0 implies that N(( t Bx : t > s)) _ s-1N((tBx: t > 1)) for all s > 0, we see that, for N E A with WN E A3, Q.E.D. supp N = {t BxN: t > 0) U (0) for some XN E Rd. Lemma 4.6.2. Let B E Aut+(R d) and let G be a measure on ARd \ (0)) such that t - G = tBG for all t > 0 and supp G = {tBxo: t > 0) U (0) for some x0 E Ind. Then G is a Levy measure, that is, f (IIxII2 A 1)G(dx) < -, if and only if xo E ker h1(B).
Proof. Let y E SB be such that y = to xo for some to > 0. Then
supp G = {t By: t > 0) U (0). For the first part of the proof, assume G is a Levy measure. Hence, by Proposition 4.3.4, G(A) _ cfoJA(tBy)t-2 dt, where c := m({y}) = G({say: s > 1}). Thus, cf 111tByIIBt_2dt 0
= fIIXIIBs 1 IIxIIBG(dx) < oo.
Let X := linB{y}, that is, the smallest B-invariant subspace containing y. Since B is invertible, BX = X so we may consider B' on X, where B'
is the restriction of B to X. By Theorem 1.2.3, X may be
decomposed into the direct sum of subspaces X1,. .. , Xk with B'X1 =
Xi and with the minimal polynomial qj of B, being a power of a polynomial which is irreducible over the reals, where B1 is the restric-
CHARACTERIZATIONS OF OPERATOR-STABLE MEASURES
207
tion of B' to Xj, 1 < j < k. Write y = El= I y, with each y, E X,. Then y, * 0 since y generates X. Clearly, tBy = Ek=,tBy,. The problem now is to compute t B, y, in terms of the eigenvalues of B, that is, of B,. Let 7,(a1,.. . , aP) denote the upper triangular matrix with a, on the principal diagonal, a2 on the super diagonal, and so on, and aP as the (1, p) entry.
First, assume that q, is linear, that is, q,(t) = t - a with a > 0. Since T,,, (a, 1, 0, . . . , 0) = aI + N, where I is the m, X m, identity matrix and N is the m, X m, matrix with all entries being zero except
for all ones on the super diagonal, and because Ni # 0 for 0 S j <m, - 1, N'"- = 0, we see that q;""(T,,, (a, 1, 0, . . . , 0)) = 0 and no polynomial of lower degree has this property. Hence, q; is the minimal polynomial of Tm (a, 1, 0, . . . , 0); consequently, there exists a basis (z 1 , ... , z,,,r) for X, such that Tm(a, 1, 0, ... , 0) is the matrix representation of Br with respect to this basis. Also, m,-1
tB, = tatN = to T, (log t)'(j1)-'N' j=0
= Tm,(ta, t" log t,...,
ta(log t)"''-'/(m, - 1)!).
Define a norm II II, on X, by IIEjajzjll,
E;la;l Therefore, IItB'y,ll; is equal to a sum of terms of the form c,t2a(log t)12 with constants c, > 0 and 0 S c2 < 2m,. - 2. Now, we may define a norm II - Ilx on X by IIEk=1x,IIx
Ek=,Ilxll, Then IIEk=1x,IIX ? 11X,ll; for 1 < r s k.
Since JoIIt,Yllat-2 dt < oo, we have JoIItBry,ll;t-2 dt < -0 for each 1 < r < k. However, the last integral is equal to the sum of terms of the form cJot2a-2(log t)b dt with c > 0 and 0 < b < 2m, - 2. Hence, each of these integrals is finite which implies that 2a - 2 > - 1, that is, a > 1/2. Therefore, X c ker h1(B). Consequently, y E ker h,(B), so we have x0 E ker h 1(B).
Second, assume q, is quadratic, that is, qr(t) = t2 - 2at + a2 + 132
with a > 0 and 6 > 0. In this case, we must extend X, and B, to their complexifications, that is, X, _= X,. + i ,. and B`(x + ix) B,x + iB,x for x c= X,. Proceeding as before, we find there exists a complex basis {z1, ... , Z2m) of X, such that the matrix representation of B; with respect to this basis is Tm,(a + i/3,1, 0, ... , 0)
0
0
TT,(a - i13,1,0,...,0)
OPERATOR-STABLE MEASURES
208
Ideas similar to those when q,. was linear lead to obtaining t2a-2(log t)b dt < oo with b >_ 0. f1 0
Therefore, as before, a > 1/2 which implies that x0 E ker h1(B). To obtain the converse, assume x0 E ker h1(B). Using the norms constructed in the necessary part of this proof, we have 1lIItRy,II t_2 dt < oo, 0
each 1 5 r s k. Hence, f 111tBy1I t_2dt < 00. 0
Consequently, f111tByIIBt-2dt < oo, 0
which implies f(IIx112 A 1)G(dx) < oo, that is, G is a Levy measure. Q.E.D. Lemma 4.63. Let B E Aut+(Rd) and let G be a Levy measure on Rd such that t - G = tBG for all t > 0. Then supp G c ker hl(B). Proof. By Lemma 4.6.1, there is a sequence (Nn) of Levy measures
such that, for all n z 1, t - Nn = t BNn for all t > 0, supp Nn = t > 0) U {0} and Nn ker h1(B) for all n >_ 1, so supp Nn U ka 1(t Bxn:
G. By Lemma 4.6.2, x,, E
ker h1(B). Since ker h1(B) is closed, (U-,,-, supp Nn) c ker h 1(B ). Let x e (U n =1 supp Nn ). Select
an open neighborhood U of x sufficiently small so that U n (U'- n1 supp Nn) = 0. Also, select U so that G(W) = 0. Then G(U) _ Jim Nn(U) = 0. Therefore, x e supp G. Consequently, we have supp G C (Unn1 suppNn)c kerh1(B).
Q.E.D.
Lemma 4.6.4. Let B E Aut+(Rd), µ = [a, R, M] be t '-stable, not necessarily full, and y [0, R, 0] be the zero-mean Gaussian compo-
CHARACTERIZATIONS OF OPERATOR-STABLE MEASURES
209
nent of µ. Then (a) supp y is a B-invariant subspace;
(b) supp y c ker g(B). P r o o f. For (a), let 71 :_ [a, 0, M], so A = y*?7. Then µ' = t6µ * S(b(t)) for all t > 0 implies that
y'*7n' = tBy*tBq*S(b(t)) for all t > 0. Since y' and t 6y are zero-mean Gaussian measures and the other factors have no Gaussian components, we see that y' = tey for all t > 0, that is, y is t 6-stable. From Theorem 1.8.20, since y has mean zero and covariance operator R, its support is given by (ker R) , so it is a subspace of R'. All this implies that
supp y = supp y' = supp t8y = t6(supp y). Let v E supp y. Then (t - 1)-'(t6 - I)v c- supp y for all t > 0. Since these vectors converge to By as t - 1, we obtain By E supp y, so supp y is B-invariant.
For (b), let V:= supp y, B, := BIV, and 1, be the identity on V. Since, for t > 0, 9'(y) = y(t'/2y), t('/2 is an exponent for y. Hence, tB y = y' = t''/2y implies t6'-''/2 E A,,(y) :_ (A E Aut(V): Ay = y). Since y is full Gaussian on V, A,,(y) = W-'6,,W for some W E Aut(V), where B,, is the group of all orthogonal transformations on V (cf. Proposition 2.3.11). Hence,
W(t6,-(,/2)W-I = tw(B,-/,/2)w-' E B v for all t > 0. Hence, t w(6' -1, /2)w
commutes with its adjoint and
tW(6,-1,/2)w-'+[w(B,-(,/2)W-'l
=I
for all t > 0. Differentiating with respect to t and evaluating at t = 1, we see that W(B1 - 1,/2)W-' is skew-symmetric. Since the eigenvalues of any skew-symmetric operator are always simple and have real
parts equal to zero, the same holds for B, - I1/2. Therefore, the eigenvalues of B, are simple and have real parts equal to 1/2, so this is also true for B. Let g, be the minimal polynomial of B,. Then the
roots of g, are simple and have real parts equal to 1/2. Thus, g, divides g. (Recall that g is that factor of the minimal polynomial of B
OPERATOR-STABLE MEASURES
210
containing all roots whose real parts are equal to 1/2.) Hence, for x E supp y, g,(B)(x) = g,(B,)(x) = 0 since g,(B,) is identically equal to zero. Thus, g(B)(x) = g,(B)(x) = 0, so x e ker g(B). Therefore, Q.E.D. supp y c ker g(B). Theorem 4.6.5 (Structural Theorem). Let µ be a full operator-stable measure with exponent B. Then the minimal polynomial of B may be factored into g - h, where the roots of g have real parts equal to 1/2 and
the roots of h have real parts strictly greater than 1/2. Furthermore, µ = µ, * µ2, where µ, is Gaussian and µ2 has no Gaussian component, supp µ, = ker g(B), lin(supp µ2) = ker h(B), and Rd = supp µ, ® lin(supp µ2). Finally, for i = 1, 2, letting B. denote the restriction of B to lin(supp µ;), we have that each µ; is a full tBi-stable measure on the space lin(supp µ). (Later, we show that the roots of g are simple; cf. Theorem 4.6.12.)
Proof. Let µ = [a, R, MI and set y := [0, R, 0], 77 that µ = y * rl * S(a). By Theorem 1.2.3, Rd
[0, 0, M], so
= kerg(B) ® ker h,(B) ® kerh2(B),
where g, h,, and h2 are as in Lemmas 4.6.1 and 4.6.2. By Lemma 4.6.3, lin(supp 77) c lin(supp M) c ker h,(B). By Lemma 4.6.4, supp y c ker g(B). Since µ is full, we have R = supp(p. * S(-a)) = supp y * 77 = supp y + supp -q
c kerg(B) ® kerh,(B) c Rd. Therefore, Rd = ker g(B) ® lcer h,(B), supp y = ker g(B), lin(supp 17) = ker h,(B), and h2 = 0. Now, represent a as a, + a2 with a, E ker g(B) and a2 E ker h,(B). Set µ, := y * S(a,) and µ2 :_ 77 * 8(a2), thereby establishing the second statement in the theorem.
Because µ is tB-stable, µi * 2 = t8µ, * t8µ2 * 6(b(t)) for some vectors b(t) and for all t > 0. Thus, µ; and t8µ, are the Gaussian components of the same infinitely divisible measure, so µ1 = t8µ * 3(c W) for some vectors c,(t) E Rd and for all t > 0. Hence, y t = t B y * S(- ta, + t Ba, + c,(t )) for all t > 0. Also, supp y' = supp y = ker g(B) and supp tey = tB(supp y) = tB(ker g(B)) = kerg(B) since kerg(B) is B-invariant. Hence, -ta, + tBa, + c,(t) E ker g(B). Since a, E ker g(B), c,(t) E ker g(B) for all t > 0. There-
211
CHARACTERIZATIONS OF OPERATOR-STABLE MEASURES
fore, 1AI is full and t B--stable on kerg(B) = supp u,. Similarly, 17' _ t871 * S(-tae + tBa2 + c2(t)) for some vectors c2(t) E Rd and for all
t > 0. By Proposition
3.4.10, lin(supp q') = lin(supp t M) = lin(supp M) = ker h,(B). Also, since tB is linear and invertible, lin(supp tB7t) = t8(lin(supp 71)) = tB(ker hl(B)) = ker hl(B). Thus, c2(t) E ker h,(B) for all t > 0, and so µ2 is full and tB2-stable on
ker hI(B) = lin(supp µ2).
Q.E.D.
The next goal of this section is to consider, for a fixed full operator-stable measure p., a characterization of the class of all possible exponents B. We denote this class by d(µ), that is,
4',(µ) = {B E Aut(R"): µ E where J(B) is the class of all tB-stable measures. We first describe some basic properties of 6' (µ), recalling that OS denotes the class of all t '-stable measures for some B. Proposition 4.6.6
(a) ForA E Aut, A.'(B) = .J'(ABA-'). (b) ForA E Aut and µ E OS, A ,(tz)A-' = S(AIL). (c) ForA E A(µ), µ E OS, and B E 6' (µ), ABA-' E 6;(µ). (d) For any vector a, d(µ) = 9' (µ * S(a)). (e) The set d(µ) is closed.
Proof. For (a) and (b), since u' = tBµ * S(b(t)) for
all
t > 0,
t ABA-'(AA) * S(Ab(t)) = At BA * S(b(t))) = (Aµ)`. Therefore, µ E
,(B) and A E Aut implies that Aµ E .J'(ABA -') and ABA -' E d' (Au). Hence, c I(ABA-') and A4' (A)A-' c d,'(Aµ). Similarly, A-'.,(ABA-') c .J'(B) and A-'ds(AA)A c d' (A). For (c), since µ = Aµ * S(a) for some vector a, we have
Aµ` = At8µ * S(Ab(t)) = tABA-'(Ap.)* 3(Ab(t)) = tABA`µ * S(Ab(t) _ tABA -'a) Hence,
µ` _ (Aµ *3(a))' = Au'* S(ta) =
tABA-'µ
* S(ta + Ab(t) -
tABA-'a)
OPERATOR-STABLE MEASURES
212
For (d), let B C 6' (µ). Then u' = tBµ * S(b(t)) implies (µ * 6(a))` _
tB(µ * S(a))* S(b(t) + to - tBa), so Be d'(µ * S(a)). Thus, 6' (µ) c 4' (µ * 6(a)). The reverse inclusion is just as easy.
For (e), let B E d(µ) be such that B,, -* B E End. Then, for all n > 1 and all t > 0, µ` = t B^µ * S(b (t )) for some vectors bn(t ). Since tBµ, we see that for each t > 0 there is a vector b(t) such that tB^µ Q.E.D. b(t) Therefore, µ` = tBµ * S(b(t)), so B C 4;(µ).
Before stating the theorem which characterizes the class 4'(µ), we recall some definitions and notation. In Section 2.3, we showed that the symmetry semigroup A(µ) :_ (A E Aut: µ = Aµ * 6(a) for some
vector a) of a full measure µ is a compact subgroup of Aut (cf. Corollary 2.3.2). In Section 1.5, the tangent space of any closed subgroup G of Aut was defined and denoted by .l(G), that is, (A E End: A = 9-(G) b,-,1(g - I) for some sequence {g,,),, ,1 in G with 0 < b,, - 0 as n - x}. Many of the results on Lie algebras collected in Chapter 1 come to play in proving the following theorem.
Theorem 4.6.7 (Structure of the Exponents). Let µ be full and operator-stable on Rd with an exponent B. Then
4;(µ) = B + Proof. We first recall many of the results used to prove Sharpe's representation theorem 4.2.12. For t > 0, let G, (A E Aut: µ` = Aµ * 8(a) for some vector a) and set G := U, > 0G,. Then G, = A(µ) is a compact subgroup of the closed group G (cf. Lemmas 4.2.7 and 4.2.8). Also, the function
L:
log(l(exp A)), where 1(C) = t
.1(G) -' F1 given by L(A)
for C E G is a continuous linear
functional (cf. Lemma 4.2.9).
Since A E 6' (µ) implies that t' c- G for all t > 0 and A is the derivative of tA at t = 1, we see that (f,(µ) c 1(G). We claim that 4' (t..) = (A E .1(G): L(A) = 1). For A E 6' (µ), in particular, we have µe = eAµ * S(b(e)), that is, eA C Ge. Hence, l(eA) = e so L(A) = 1. Conversely, let A E 9-(G) with L(A) = 1. By the linearity of L, L((log t)A) = log t for all t > 0. Thus, l(tA) = t, so to E G, for all t > 0, that is, A E d (µ). Therefore, 4,(µ) = (A E 1(G): L(A) = 1). By Lemma 4.2.10, .1(A(µ)) = ker L. Now, let B be any exponent of A. Then B E .1(G) and L(B) = 1. For any other exponent C of µ, we have L(B - C) = 0, so B - C C kerL = 97(A(µ)). Hence, 6' %u) c B + ,7"(A(µ)). Finally, let D E
CHARACTERIZATIONS OF OPERATOR-STABLE MEASURES
213
5"(A(µ)). Then B + D (=- .T(G) since .T(G) is a vector space, and L(B + D) = 1. Thus, B + D E 6' (µ), so 4:(µ) B + .91(A(µ)). Q.E.D.
Corollary 4.6.8. Let µ be full and operator-stable on Rd. Then µ has exactly one exponent if and only if A(µ) is finite.
Proof. From Theorem 4.6.6, we have that µ has exactly one exponent if and only if .9"(A(µ)) = (0). From Section 1.5, the image of .91(A(µ)) under the exponential mapping contains an open neighborhood U, of the identity I in A(µ). Hence, .91(A(µ)) = (0) if and only if
the one-point set (I) is open in A(µ). If .9(A(µ)) = (0), then exp(J-(A(µ))) = {I) = U, is an open set in A(µ). Because A(µ) is a group, each singleton set is open in A(µ), and the compactness of A(µ) implies it is a finite group. Conversely, if A(µ) is finite, then (I) is an open neighborhood of the identity, so .91(A(µ)) = (0). Q.E.D.
Let µ be full and operator-stable on Rd. Then the (µ) is affine, that is, for all a E 68' and B1, B2 E o(µ), we have
Corollary 4.6.9. set
aB, + (1 - a)B2 E6°(µ).
Proof Fix B E 6°(µ) so that 6' (µ) = B + .9(A(µ)). Then, for B1, B2
there are C,, C2 E .91(A(µ)) such that B; = B + C,,
i = 1, 2. For any a E R', aB, + (1 - a)B2 = B + aC, + (1 - a)C2. Since .91(A(µ)) is a vector space, we have aC, + (1 - a)C2 E 9(A(µ)), so aB, + (1 - a)B2 E (µ) Q.E.D. For a Gaussian measure µ, the class d(µ) of its exponents has a simple description given by the following theorem. Theorem 4.6.10. Let µ be a full Gaussian measure on Rd with covariance operator R. Then
d"(µ) =
I 2
+ S-'.'S,
where S is the unique positive-definite self-adjoint square root of R-'
and X is the class of all skew-symmetric linear operators on Rd. Conversely, if 6;(µ) = 1/2 + T-'JKT for some positive-definite selfadjoint linear operator T, then R = AT-2 for some A > 0.
OPERATOR-STABLE MEASURES
214
Proof. By Proposition 4.6.6(d), we may assume that µ has mean zero. Consider the case when R = I. We first show that d' (,u) ;? 1/2 + .1E', so let K E A. Clearly, 1/2 E e(µ); consequently, t!/2+Kµ = tKte for all t > 0. However,
(t"p!)^(y) = exp{-t(t
,
K*y)/2} = exp(-t(y,y)/2)
since K + K* = 0. Thus t112+Kµ = Ae for all t > 0, so 1/2 + K E d;(p.), that is, o(µ) D 1/2 + .X' Now, let B (=- CA). By Proposition 4.3.2, t I = t Bt B* for all t > 0, so t B -!/2t B* _'/2 = I. Taking the
derivative at t = 1, we obtain (B - 1/2) + (B* - 1/2) = 0, that is, B - 1/2 E .1l', so d(µ) c I/2 + X. Consequently, 4' (µ) = 1/2 + , when R = I.
'
For general R,
(Sµ)^(y) = exp{-(RSy,Sy)/2} = exp{-(y,y)/2}, that is, the covariance operator of Sµ is the identity, 1. Hence, 6' (Sµ) = 1/2 + X. By Proposition 4.6.6(b), 6' (µ) = 1/2 + S-'.3l'S. For the converse, note that the covariance operator R of µ and the covariance operator U of Tµ are related by U = TRT, since T is self-adjoint. Also, by Proposition 4.6.6(b), o(Tu) = Td (A)T-' _ 1/2 + X. Hence, for all K E J Y and for all t > 0,
t. U= t 1/2+KUt!/2+K*
or Ut K= tKU.
Thus, U commutes with everything in exp(.'). Since exp(.1E') contains all rotations on Rd (cf. Section 1.5), we see that U commutes with every rotation. Consequently, we have that U = AI for some real A (cf. page 180). Since U is positive-definite, A must be positive. Therefore, TRT = Al or R = AT-2. Q.E.D. An easy corollary to the theorem is the following. Corollary 4.6.11. Let µ be a full Gaussian measure on Rd with covariance operator R and let B be an invertible linear operator on Rd. Then B is an exponent for µ if and only if S(B - I/2)S-' is skew-symmetric, where S is the positive-definite self-adjoint square root of R-'.
Earlier (cf. Corollary 4.2.13), we showed that if B is an exponent for a full measure, then all of the eigenvalues of B have positive real
CHARACTERIZATIONS OF OPERATOR-STABLE MEASURES
215
parts. Then in Theorem 4.6.5 we established that those eigenvalues
must have real parts greater than or equal to 1/2. This naturally raises the question of whether or not every B E Aut(Rd), whose eigenvalues have real parts greater than or equal to 1/2, is an exponent for some full measure? The answer is in the affirmative with the additional requirement that the roots of the minimal polynomial
of B, with real parts equal to 1/2, are simple. Theorem 4.6.12 (Spectral Characterization of Exponents). A necessary and sufficient condition that a linear operator B on Rd be an exponent for some full operator-stable measure is that all of the roots of the minimal polynomial of B have real parts greater than or equal to 1 /2 and those with real parts equal to 1 /2 are simple roots.
Proof. Necessity. Assume B E 4' (µ) for some full µ. As before, factor the minimal polynomial of B into the product g h, where the roots of g all have real parts equal to 1/2 and those of h have real parts strictly greater than 1/2 (cf. Theorem 4.6.3). It remains only to show that the roots of g are simple. Let µ, be the Gaussian compo-
nent of p and let B1 be the restriction of B to ker g(B). Apply Corollary 4.6.11 to µ, and B, on the space kerg(B). Then S(B1 1,/2)S-' is skew-symmetric for some invertible linear operator S on kerg(B), where I, is the identity on kerg(B). Hence, the minimal polynomial g, of B, has only simple roots with real parts equal to 1/2 (cf page 209). But, g, = g. Sufficiency. Let g and h be as usual. By Theorem 1.2.3, decompose R1 into the direct sum of subspaces V,, ... , V, such that BV = Vi and the minimal polynomial of the restriction of B to Y is a power of a real-irreducible polynomial, for j = 1, ... , r. Number the subspaces V so that the eigenvalues in V1,. .. , Vk have real parts greater 1 /2 and those in Vk+,,... , Vr have real parts equal to 1/2. Set Vo :_ Vk +, ® ® Vr. Let S1 .= SB n V for j = 1,. .. , k, where SB is the unit sphere under the norm II IIB. Let m, be a Borel measure on SB with supp mf = Si, 1 < j < k. Hence, linB(supp m) c Y, since Y is B-invariant. Let x c- V. Then x/II xII B E SB n V = supp m,, which implies that x E linB(supp m). Therefore, lin8(supp m .) = V, for j = 1, ... , k. Define Mi on the Borel subsets of R" \ {Of by M1(A) JsgfoIA(sBx)s_2
dsmj(dx). Then M1 is a Levy measure and t
M1 = t 'M. for all t > 0 (cf. Proposition 4.3.4 and Lemma 4.6.1). Let vj := f0, 0, M1]. From Corollary 4.3.5, we see that lin(supp vj) =
OPERATOR-STABLE MEASURES
216
lin(supp M) = linB(supp M) = V, so v
vk = [0, 0, M ], with M
VI*
v, is full on V. Therefore, Ek= I M., is t B-stable and full
onVl®...9Vk.
Now, to construct a Gaussian measure y which is full on V0 and t B-stable, let B0 denote the restriction of B to V0. Since the minimal
polynomial of B. has only simple roots with real parts equal to 1/2, there is some basis for V. such that the matrix representation of B0 with respect to this basis is block-diagonal, with the blocks being either 1 X 1 matrices with entry 1/2 or 2 x 2 matrices of the form
with P
0. Let Io be the identity on V0. Then B0 - 1,/2 is skew-
symmetric, so tBO-b0/2t Bo -`O/2 = Io
or
t - I0 = tB0IQt B°
By Proposition 4.3.2 and Theorem 1.8.20, y [0, 10, 01 is t '-stable, Gaussian with zero mean, and full on Vo with covariance operator J. Set µ = y * v. Then µ is full on Rd and t '-stable. Q.E.D. We conclude this section with a necessary and sufficient condition for the compatibility of a Levy measure M and an exponent B. Theorem 4.6.13. Let M be a Levy measure on W' and B E Aut with its minimal polynomial of the form g - h, where the roots of g have
real parts equal to 1/2 and are simple, unless g is identically one, and the roots of h have real parts greater than 1/2. Then there exists a full t B-stable measure µ on Rd whose Levy measure is M if and only if both
t - M = tBM for all t > 0 and lin(supp M) = ker h(B). Proof. Assume M is the Levy measure of the full t B-stable measure ,u. By Proposition 4.3.2, t - M = t 6M for all t > 0. By Theorem 4.6.5,
there is a measure 1L2 having no Gaussian component such that A = N'1 * µ2, lin(supp µ2) = ker h(B) and M is the Levy measure of µ2. Therefore, ker h(B) = lin(supp µ2) = lin(supp M). Conversely, let µ2 [0, 0, M]. Then µ2 is full and operator-stable
on ker h(B) with exponent B2
BIkerh(B). As in the proof of
217
CHARACTERIZATIONS OF OPERATOR-STABLE MEASURES
Theorem 4.6.12, construct a Gaussian measure µI on Rd such" that µI
is full on ker g(B), BI Biker g(B) is an exponent for A,, and A `= Al * A2 is full on Rd, tB-stable, and with Levy measure M.
Q.E.D.
In Theorem 4.6.5, we saw that a full t B-stable measure µ has a B2-stable Poisson compot B1-stable Gaussian component A, and a t nent µ21 where B. = BIV,Z, i = 1, 2, even though µI and µ2 are not full. We will need a condition on a projection P on Rd so that when µ is t B-stable, not necessarily full on Rd, Pµ is t BP-stable. Proposition 4.6.14. Let µ be t B-stable on Rd and let P be a projection on Rd which commutes with B. Then Pµ is t BP-stable on fed.
Proof. Using the fact that P and B commute, we see that
(Ps)` = P(tBA* S(b(t))) = (tBP)A * b(P(b(t))) = tB'(Pµ)* S(P(b(t))). Q.E.D.
Proposition 4.6.15. Let µ be full and t B-stable on Rd and let P be either the projection onto V2 with ker P = VI or the projection onto VI with ker P = V2. Then P and B commute.
Proof From Theorem 4.6.5, we know that both VI and V2 are B-invariant subspaces. Let P be the projection onto V2 with ker P = V1. Since B(I - P) y E VI, we see that PBy = PBPy + PB(I - P) y = PBPy = BPy for all y. Hence, PB = BP. The other case is as simple. Q.E.D.
Finally, we will need a partial converse to Proposition 4.6.14. Proposition 4.6.16.
Let V be a proper subspace of Rd and let v be
a full measure on V. Let µ(F) v(F n V) for F E .(R'). If v is t C-stable on V for some C E Aut(V), then p' = t'A * S(c(t )) for all t > 0, where D is any linear operator on R' whose restriction to V is C,
and c(t) E V.
OPERATOR-STABLE MEASURES
218
Proof. Let P be the orthogonal projection of Rd onto V. Then, for any t> O and y E Rd, r
X(y) = (fV
ei 0, by Proposition 4.3.2, p. [0, R, M] is tB-stable. Clearly, µ is full on 682, supp A, = V!, and lin(supp µ2) = V2. However, R(x,x)* =x. 4.7 COMMUTING EXPONENTS AND OTHER NORMS
In many problems, it would be an advantage to know that whenever B E o(µ) and A E A(µ) we have tBA = AtB for all t > 0, or, equiva-
lently, BA = AB. That this is not necessarily true is seen by the following example.
Example. Let µ be the symmetric Cauchy measure on R2, that is, µ(y) = e-11"11 for y E 682. Clearly, A(µ) is the full orthogonal group.
Also, (t'µ)^ (y) = e-Jliyu = µ`(y) for all t > 0 and for all y E 682. Consequently, I E d',,(A). Therefore, by Theorem 4.6.7, d(µ) = I + .X, where . ' is the class of all skew-symmetric linear operators on 12, since Y(®) = .Z (cf. Section 1.5). Hence,
B:=(_11
1) E e(µ)-
A.=(0
?)EA,
1
But, for
we have AB * BA. Note that in this example there does exist an exponent, namely the identity, which commutes with A(µ). This raises the question: when
does such an exponent exist? We show that the answer to this question is: always! This prompts other problems.
1. Describe the class of all exponents of µ which commute with A(µ). 2. Find a necessary and sufficient condition so that all exponents of µ commute with A(µ). 3. Knowing that such "commuting" exponents always exist, what can one say about the measure mB on SB given in the represen-
OPERATOR-STABLE MEASURES
220
tation of Theorem 4.3.7 (cf. Proposition 4.3.4), that is, if µ = [a, R, M] and B E o(µ), then
M(A) = f f
WIA(sBx)s_2
dsmB(dx)
for A E _q(Rd \ (0)).
sB 0
We now proceed to solve these problems. Let µ be operator-stable on Rd and denote the class of "commuting" exponents by 9' S(10, that is,
d%(µ)
(B E d :(A): BA = AB for all A E A(µ)).
We call each B E 6c,(µ) a commuting exponent of A. Theorem 4.7.1. Let µ be a full and operator stable measure on OBd. Then 6s(µ) is nonempty.
Proof. Since o(µ) * 0 by Theorem 4.2.12, we may select B E 6's(µ). By Corollary 2.3.2, A(µ) is a compact group, so let H denote the normalized Hart measure on A(µ). Define Bc
f ABA-'H(dA). A(µ)
Note that, for all A E A(µ), ABA ' E 4'.,(µ) by Proposition 4.6.6(c). Also, 4' (µ) is closed and affine [cf. Proposition 4.6.6(e) and Corollary 4.6.9]. Thus, Bc E d's(µ). Using the invariance property of the Harr measure, we obtain, for D E A(µ),
DBcD-' = f (DA)B(DA)-'H(dA) = B, A(µ)
Therefore, DBc = BED, which implies Bc e s(µ).
Q.E.D.
Corollary 4.7.2. When A(µ) is finite, the unique exponent B of the full operator stable µ is commuting.
This corollary follows from Corollary 4.6.8 and Theorem 4.7.1.
Lemma 4.73. Let µ be a full and operator-stable measure and let W E Aut. Then 4'cs(Wµ) = Wd ,(µ)W-
221
COMMUTING EXPONENTS AND OTHER NORMS
Proof. Since
s(µ) = gy(p) n (B: BA = AB for all A E A(µ)), from Proposition 4.6.6(b) and Lemma 2.3.3, we have
4; (Wµ) n ( WBW-': BA = AB for all A E A(µ))
_ 4' (Wµ) n (C: CD = DC for all D E A(WA)) (6S(WA).
Q.E.D.
Later, we shall often use the following simple observation.
Lemma 4.7.4. Let µ be a full and operator-stable measure. If then B and B, commute and exp s(B B E r(µ) and B., E
Bd E A(µ) for every s e R.
Proof. Since B - B, e ,7-(A(µ)) by Theorem 4.6.7, B - B, is the
limit of (D - I)/d for some D (=- A(µ) and 0 < d - 0. Since B, commutes with each D, B, must commute with B - B, which is equivalent to B, commutes with B. We have that exp s(B - B,) E Q.E.D. A(µ) for every s e W. Thus, a commuting exponent is an exponent which commutes both with the symmetry group and with the class of all exponents of A. Now that we know that e,(µ) is nonempty, we wish to characterize
it in a manner similar to the characterization of 4(µ) in Theorem 4.6.7. Since G°S(µ) concerns operators which commute with A(µ), it is natural to expect that its characterization involves those operators in
A(µ) which commute with all of A(µ), the so-called center of the group A(µ). We denote the center of A(µ) by Cent(A(µ)), that is, Cent(A(µ)) := (C E A(µ): CA = AC for all A E A(µ)}. When µ is full, it is easy to see that Cent(A(µ)) is a compact subgroup of the compact group A(µ). Hence, the tangent space ,T(Cent(A(A))) exists and is a subspace of J"(A(µ)). Theorem 4.7.5.
Let µ be a full and operator-stable measure on F
Then, for any B E
6',S(µ) = B + .T(Cent(A(µ))).
.
OPERATOR-STABLE MEASURES
222
Proof. For any B, E Q5(µ), exp s(B, - B) E A(µ) for all s E RI (cf. Lemma 4.7.4). Also, since B, - B commutes with A(µ), for A E A(µ) and s c- 181, we have A(exp s(B, - B))A-1 = exp sA(B, B) A` = exp s(B,, - B). Consequently, exp s(B, - B) E Cent(A(µ)) for all s. Thus, B, - B E .T(Cent(A(µ))). Therefore, 6C5(µ) c B + Y(Cent(A(p.))).
To obtain the reverse inclusion, let B' - B + D for some D E ,T(Cent(A(µ))). Since ,T(Cent(A(A))) c .T(A(µ)), we have B' E d(µ) (cf. Theorem 4.6.7). Also, for all s, exp s(B' - B) E Cent(A(µ)). Hence, exp s(B' - B) commutes with A(µ), and consequently B' - B commutes with A(µ). Since B commutes with A(µ) by definition, we have that B' also commutes with A(µ). Therefore, B' E d' 5(µ). Q.E.D.
The following corollary provides an answer to: when is every exponent commuting? Corollary 4.7.6.
Let µ be a full and operator-stable measure. Then is commuting if and only if .9 (A(A)) _
every exponent of p. .T(Cent(A(p.))).
In Section 3.4, a very useful norm was defined by iixllB := f'0Ile-`Bxll dt = 0
f'11sBx11s-1 ds,
0
where all of B's eigenvalues have positive real parts. This norm was used to solve the equation t M = tBM for all t > 0. The solution was that M must be of the form
M(A) = fsB f IltBxlIBt-2dtmB(dx), where SB
{x E Rd: Ilxlis = 11 and mB is a Bore] measure on SB. A
disadvantage of this representation is that the sphere S. and the measure mB depend on the exponent B and may not be invariant under the action of the symmetry group of [a, R, MI. The knowledge that there is always a commuting exponent, when [a, R, MI is full, allows us to define another norm such that the unit sphere under this new norm does not depend on the exponent and is invariant under the action of the symmetry group. A unit sphere in this norm can be used to solve t M = t BM, t > 0, obtaining a measure which also does not
COMMUTING EXPONENTS AND OTHER NORMS
223
depend on the exponent and is invariant under the action of the symmetry group.
Let B E 61,(A) with µ full. For x E Rd, define
Jf
IIXIIµ,B
IIIAtexllt-I
A(µ) 0
dtH(dA),
(4.7.1)
where H is the normalized Haar measure on the compact group A(µ). Since IIxIIµ,B < oo for all x E Rd, it is obviously a norm on Rd (cf. Section 3.4). For µ full, the norm II ' 11,,,B does not depend on the
exponent B of µ. Simply note that for B E o (µ) and B' E o(µ), B and B' - B commute and exp(s(B' - B)) E A(µ) for all s E R' (cf. Lemma 4.7.4). Therefore, by the invariance of the Haar measure, J1f IIxIIN.,B' = 0
IIAtB'-BtBxllt-'H(dA)dt
A(µ)
= Ilxllµ,B.
Hence, we may define the norm II ' Ilµ for A full and operator-stable by Ilxllµ := IIXII, ,B
(4.7.2)
for some B E 4 (µ). Letting B E 6' S(µ), we see that the norm II ' Ilµ is invariant under A(µ), that is, for A E A(µ), IIAxlIN, = Ilxllµ for all x. Let Su denote the unit sphere with respect to this norm, that is, S" := {x E Lid: Ilxllµ = 1).
In Section 3.4, we defined the norm II ' IIQ for any Q such that
exp(-tQ)-->0as t ->ooby IIXIIQ = f IIItQxllt-' dt. 0
Using B E 60,(µ) for Q, we see that
Ilxllµ = f
IIAxll BH(dA).
(4.7.3)
MIL)
Similar to the function cQ defined in Section 3.4, for '.c full and operator-stable with B E d' (µ), we define 'Ye: SA x (0, 00)
Rd \ (0)
by
'Y'(u, t) := teu.
(4.7.4)
OPERATOR-STABLE MEASURES
224
Proposition 4.7.7. Let it be a full operator-stable measure and let
B E 4s(µ). (a) W. is a homeomorphism between Sµ x (0, co) and Rd \ (0}.
(b) For each nonzero vector x, the function 0 < s .
II sBxiI µ
is
strictly increasing.
(c) For E E A(SA), G E 2((0, -)), and s > 0,
sB*B(E x G) _B(E x sG). (d) For E E Y(S.) and G E 2((0, co)), we have, for any A E A(µ) which commutes with B,
A*B(E X G) = IYB(AE X G). (e) For Bc E &IcS(µ), u E Sµ, and t > 0, we have
TB(u, t) = tB-B, , (u, t).
Proof. The proofs of (a), (b), and (c) are similar to the proof of Proposition 3.4.3, so they are omitted here. For (d), since S,, is A(µ)-invariant,
For (e), by Lemma 4.7.4, B and B - Bc commute so 418(u, t) = t B-B°t B`u = t B-B` ",(u, t).
Q.E.D.
Define functions uB: Rd \ (0) - S. and TB: Rd \ (0) - (0, oo) by
(uB(x),TB(x)) °_ TB'(x), that is, uB(x) is that unique point on the sphere Sµ which is on the orbit tBx and TB(x) is the "time" to reach x from the origin traveling on the orbit t Bx. This gives two families of functions indexed by the class of exponents of µ. However, the family of "times" actually consists of exactly one function. Also, the uB's are closely related.
Proposition 4.7.8. Let µ be a full operator-stable measure. The functions TB for B E 6(µ) are all the same and may be simply denoted
COMMUTING EXPONENTS AND OTHER NORMS
225
by T. The functions uB satisfy
UB`(X) = (T(X))B-BUB(x)
for all B E 4' (µ) and for all Bc E ec,(µ).
Proof. For each x E Rd \ (0), we have the unique representations
x = rew and x = s'rv for some r, s > 0 and w, v, e S. By Lemma 4.7.4, Bc and B - Bc commute and t B-B E A(µ) for all t > 0. Hence, rBw = rB,(rB-Bcw) = sBcv and rB-B,w E St,,, since Sµ is A(µ)-invariant. Because APB, is a homeomorphism, s = r and v = r B-Bw. Q.E.D. Proposition 4.7.7 gives the properties of 'APB which are analogous to those of 4)Q in Section 3.4. Those were the properties used in Section
4.3 to solve t M = tBM in terms of the norm II {{B and (DB defined on SB. Now, we may solve the equality in terms of the new norm and new sphere S. The advantage is that the sphere used in the integral representation is A(µ)-invariant and does not depend on the exponent. It would appear that the mixing measure does depend on the exponent; however, we will show in Proposition 4.7.10 that it does not.
Proposition 4.7.9. Let µ = [a, R, MI be full and operator-stable with an exponent B. Then, for all E E 2(Rd \ (0)),
M(E) = sµf l
IE(sBx)s-z dsrB(dx)
for the finite measure mB given by rnB(F) := M(NPB(F X [1,00)))
for F E a(Sµ), and supp M = u (tB(supp tB)) V (0). t>o
Also, for M satisfying the above, we have t M = t BM for all t > 0. The proof of this proposition is similar to the proofs of Proposition 4.3.4 and Corollary 4.3.5.
OPERATOR-STABLE MEASURES
226
Proposition 4.7.10. The mixing measure mB in Proposition 4.7.9 does not depend on the choice of the exponent B. Proof
Step 1. For A E A(µ) commuting with B E d' (µ), we have AAB =
mB. Let F E q(S,,). Then, noting that AM = M and A-'(F) c SN,,
1, u e F))
(AtR)(F) = M({t BA-'u: t
= (AM)((t8u: t
1, u (=- F)) = mB(F).
Step 2. For B E 4;(A) and Bc E d (µ), we have mB = ABc. Note that, for any exponent B,
M(E) = 0ri sµIE(t x)t
mB(dr) dt
J0W mB((t-BE) n Sµ)t-2 dt
Also, for the sets of the form T = (sBx: s >_ 1, x e F) = 4rB(F x [1, oo)) for some F E ,q(S.), we have (t-BT) n SE
0 if t < 1, if t _
1.
Furthermore, by Proposition 4.7.7, Lemma 4.7.4, and Step 1, we obtain n Sµ) =
for B E
in8'(tB-B,((t-BT)
n Sµ)) = tB'((t BT) n Sµ)
(µ). Hence,
thB(F) = M(T) = (o
mB((t-B'T)
n
Sµ)t-2dt
W
= f0
AB.((t-B T) n
Sµ)t-2 dt
= f thB,(F)t-2 dt = A,(F)
for all F E 9(Sµ).
Q.E.D.
227
COMMUTING EXPONENTS AND OTHER NORMS
Corollary 4.7.11. Let µ be a full operator-stable measure with the mixing measure m. Then, for all A E A(µ), we have Am = m (cf. Step 1 of the proof of Proposition 4.7.10).
In Theorem 4.3.7, the sphere SB and the measure m, which may depend on B, can be replaced by SK and m, respecRemark 4.7.12.
tively.
A partial converse of this remark is given in the following proposition.
Proposition 4.7.13. Let µ = [a, R, M ] be a full operator-stable measure with mixing measure m on S. Assume A E Aut is such that (Afin)(E) = m(E) for all Borel E c supp fin. If A commutes with some exponent B of µ, and if either R = 0 or R = ARA*, then A E A(1.).
Proof. Note that Am = m on supp m implies A -'(supp m) supp m and A(supp A) Q supp in. Hence, A(supp m) = supp A. For
FE6g(ffd\{0}),
(AM)(F) = fsupp thf0IA-'F(tex)t-Zdtm(dx) o0
(
= f uppmf0
IF(tBAX)t-2 dtm(dx) 00jF(tBX)t-2
fm)f0
dt(Am)(dx)
A(supD
= M(F).
Q.E.D.
In Proposition 4.7.13, it is not possible to remove the assumption that A commutes with some exponent of A. Consider the following example. Example 4.7.14.
Let d = 2 and B = diag(1, 2). For F E .(R2 \ {0}), set 00
M(F) = fv
IF(tB(1))t-2 dt.
Then supp M = { (s2) : s 0) is a half-parabola and, for s > 0, s - M = sBM. Let µ [0, 0, M]. Since lin(supp M) = 682, µ is full. Hence, µ is a full operator-stable measure with an exponent B, since lin(supp µ) = lin(supp M) = 682.
OPERATOR-STABLE MEASURES
228
To find A(µ), note that A E A(µ) if and only if AM = M. However, AM = M implies that A(supp M) = supp M. The only A E Aut taking a half-parabola onto itself is the identity. Therefore, A(µ) = (I), so B = diag(1, 2) is the unique exponent of µ. Hence, for x = (X Z), IIxIIN,=IIXIIB (x2)t-1 dt
t2
f0'(0
fo1(xi + x2t2)1"2 dt
if x2=0,
IIxII IIxII
if x1=0,
2
IIxII
2
+
x; 21x21
log(
IX21 + IIxII 1 Ix1I
if x, ' 0, x2 * 0.
)
The intersection of supp M and Sµ is the singleton u _{(0:897)) Define th on . (Sµ) by 1
th(E) =
0.893 0
if u c E, if u
E.
Then fs
f IF(tBx)t-2dtm(dx) K
0
= m({u}) fo IF(tBU)t-2 dt
f
Co
I,(tB()/t-2dt = M(F),
0
that is, in is the mixing measure for µ. Hence, all A E Aut(R2) with u
for a fixed point satisfies (At)(E) = m(E) for all E c supp m = (u). But, the only A E A(µ) is A = I.
229
ELLIPTICALLY SYMMETRIC OPERATOR-STABLE MEASURES
Instead of the norm II Ilµ given in (4.7.1), one can use the norm induced by the inner product Remark 4.7.15.
fSµ 0f1(AtBX, AtBy)t-' dtH(dA),
(x, y)I
where , ) is the usual inner product on R". Another possible norm uses (.
/p
IIXIIP,B :_
(f'jjt`xjjPt-' dt)
,
p >_ 1.
0
In particular, when p = 2, this norm is easy to compute in the previous example.
4.8 ELLIPTICALLY SYMMETRIC OPERATOR-STABLE MEASURES
For a full Gaussian measure µ on ll with covariance operator the identity, the level curves of its density are circles and its symmetry group A(µ) is the group d of all orthogonal operators. For a general invertible covariance operator R, the level curves become ellipses and the symmetry group becomes S-'®S, where S is the positive-definite
self-adjoint square root of R-' (cf. Proposition 2.3.11 and Theorem 4.6.10). This suggests the following definition. A measure µ on lid is said to be elliptically symmetric provided its symmetry group A(µ) is W-conjugate to 6 for some W E Aut(l '), that is, A(µ) = W-'®W. The characteristic function of a full operator-stable elliptically symmetric measure µ has a particularly simple form which shows that µ is either purely Gaussian or has no Gaussian component. This can also be obtained from the special form of the class of its exponents, 6s(µ). Theorem 4.8.1.
Let µ be full and operator-stable on l
and ellipti-
cally symmetric, that is, A(µ) = W-'®W for some W E Aut. Then there is c z 1/2 such that d" (µ) = ci + W-',%'W, where
.
' is the class of all skew-symmetric operators. Furthermore, µ is
purely Gaussian if and only if c = 1/2; and µ has no Gaussian component if and only if c > 1/2.
OPERATOR-STABLE MEASURES
230
Proof. Assume W = I. Select B E d' ,(µ). Then B commutes with all of A(µ) = B. Hence, A*B* = B*A* for all A E ', so B* also commutes with all of '. Consequently, AB*B = B*BA for all A E
',
that is, B*B commutes with all of B. By Schur's lemma, B*B = c11 for some constant c1. Hence, B and B* commute and both B and B* commute with all of 6. By a corollary of Schur's lemma, B = c1 for (cf. some constant c. By Theorem 4.6.12, c >_ 1/2. Since ,T(B) Section 1.5), by Theorem 4.6.7, d(µ) = c1 + 3' when W = I. For general W, A(µ) = W-' 'W implies that A(Wp.) = B, so by Proposition 4.6.6(b) we have that 4,(Wµ) = c1 + .7£' which implies
4"(µ)=cI+W-k'W.
The last statement of the theorem follows from Theorem 4.6.5. Q.E.D.
Now, we proceed to the characterization of elliptically symmetric operator-stable measures in terms of characteristic functions. We begin with two auxiliary lemmas. Lemma 4.8.2.
If A(µ) = B, then
(a) Ilxll = cllxll,h for some c
1/2;
(b) S, =Sc:= (xEIRd:llxll=c); (c) when c > 1/2 and p. = [a, 0, M], there is a constant k > 0 such that, for all M-integrable functions f,
fa d
iof(x)M(dx) = kfs,fo f(tx)t-(1+1/`>dtm,(dx),
where S, is the usual unit sphere in Rd and m, is a normalized (d - 1)-dimensional Hausdorff measure on _,,11 [cf. Billingsley (1986), Section 19]. Proof
(a) By Theorem 4.8.1, cI E d(µ) for some c >_ 1/2. Since Ilxllci = Iolls`xlls-' ds = c-'IIxII, we have
Ilxll, = f IIAxiIciH(dA) = c-'IIxII.
231
ELLIPTICALLY SYMMETRIC OPERATOR-STABLE MEASURES
(b) This follows immediately from (a).
(c) Let rn be the measure on Su = Sc given in Proposition 4.7.9. By Corollary 4.7.11, Am = m for all A E A(µ) = d. There-
fore, ?h = k, me for some constant k, > 0, where me is a normalized Haar measure on Sc. Hence, using mc(cA) _ cd-1m,(A) for A c S,,
f
J\co
f (x)M(dx) =
k1f f
f(s`x)s-2 dsmc(ds)
=
k,cd-Istf (o
=
k,cd-z+1/c f f
f(s`cx)s-2dsm,(dx) f(tx)t-
s, 0
dtm,(dx).
Q.E.D. Lemma 4.8.3.
fs,f-
For c > 1/2 and y c- pd, we have
(eut - 1)t-(1+1/c)dtm,(dx). 0
Let k, := jo(e't - 1)t- dt. Since c > 1, we have Ik11 < oo, and by the Cauchy theorem, k, = e-',/2cjo(e-t - 1)t- dt, so Re(k1) < 0. Hence, f °°(eit 0
- 1)t-(1+1/c) dt =
k1(( y, x))
1/c k1I(y,x)I1/c
if (y, x) > 0,
if(y,x) 0} and Si
{x E S,: (Y, x) < 0}.
y/II Y I I, we have
T(y) = IIYII'/`(k, f ((y', x))'Icm,(dx) + 11f I (y', x> I'/`m,(dx)). s;
Si
But, by the tinvariance of m,, these two integrals are the same with a common positive value independent of y' E S1. Therefore, T(y) = -RIIYII'I`, where 0 < f - (Re(k,)) fs Ku, x)I11`m,(dx) is constant for any u E S,. Second, c < 1. Since 1/2 < c < 1, the integral f00
(1 + t2)-'t2-11cdt < oo.
Hence, T(y) may be written as fW(eil
- 1 - it(y,x))t-(,+,/c)dtm,(dx)
T(y) = fSi 0
+ if f
0(
S, 0
dtm,(dx).
y, x)(1 +
By the Fubini theorem and the ezinvariance of m the last integral is equal to zero. Note that, since 1/2 < c < 1,
k2 := ff (e" - 1 - it)t-sin t t
z
00
1
dt t(1 + t2)
dt + J E
sin t tz
z t(1 + t)
dt)
_ (y,x)(-log(y,x) + k3), since
lim f
CIO E
6a sin t
tz
dt = loglal
and
sin t
k3 :_
1 dt
Jo
t2
t(1 +
2)
is a finite real constant. Similarly, when ( y, x) < 0, the second integral is equal to (y, x)(- log 1( y, x) I + k 3). Hence,
T(y) = Js, - 2I(y,x)I -i(y,x)logl(y,x)I +ik3(y,x))ml(dx). The last two integrals are both equal to zero, so T(y) = -QIIyII, where 0 < (3 :_ (Tr/2) js,l(u, x)Imi(dx) is constant for any u E S1. Q.E.D.
OPERATOR-STABLE MEASURES
234
Theorem 4.8.4. A measure µ on Rd is full, elliptically symmetric, and operator-stable if and only if
ii(Y) = exp{i(a, y) - /3IIW*-'yII°} for all y e Re, for some W E Aut, a E Rd, )3 > 0, and a E (0, 21; in which case, A(µ) = WOW-'.
Proof. By Theorem 4.8.1, cI E 6,(µ); and µ is purely Gaussian when c = 1/2, and µ has no Gaussian component when c > 1/2. Assume c = 1/2. Then µ is Gaussian and A(WA) = O. Hence, exp{i(b, y) - (1/2)/3'11Y112) for some /3' > 0, because the
covariance operator of Wµ commutes with all of O. Therefore, µ(y) = exp(i(W-'b, y) _,811W* IY112). Assume c > 1/2 and A(µ) = O. By Lemmas 4.8.2 and 4.8.3, µ(y) = exp(i (a, y) - p II Y II'/'`}, for some a E W and /3 > 0. Let a 11c,
so0- 1/2, was obtained by applying Schur's lemma and one of its corollaries via the orthogonal group. However, by using the full force of Schur's lemma, the assumption of elliptically symmetric may be slightly relaxed. First, let us note the following simple observation.
Remark 4.8.5. Let A, B E Aut be W-conjugate, that is, A = W-'BW, with W E Aut. Then a subspace V of Rd is A-invariant if and only if W(V) is B-invariant. Furthermore, (0) and ld are the only A-invariant subspaces if and only if they are only B-invariant subspaces.
Proof. Note that AV = V is equivalent to B(W(V)) = W(V). Also, W((0)) = (0) and W(Rd) _ Rd. Q.E.D. Theorem 4.8.6. Assume that µ is a full operator-stable measure and the only subspaces of 0 d which are A(p.)-invariant are (0) and Rd. Then
there are c >- 1/2 and a positive-definite self-adjoint W such that, for
ELLIPTICALLY SYMMETRIC OPERATOR-STABLE MEASURES
235
each B E C (A), one can find a skew symmetric K8 such that
B = cf + W-'KBW. Furthermore, either K. = 0 or KB = - yB I for some y8 > 0.
Proof. Since A(µ) is compact, A(µ) = W00W-' for some closed subgroup ®0 of the orthogonal group and for some positive-definite self-adjoint W (cf. Corollary 2.4.2). By Remark 4.8.5, the only 0, in-
and set variant subspaces of Rd are (0) and Rd. Let B E W-'BW. Then Bo E 6' ,(W-'µ) and A(W-'µ) = ®0 (cf. Lem-
Bo
mas 4.7.3 and 2.2.6, respectively). Let BI (B0 + B, *)/2 and B2 (B0 - B0 *)12. Then B0 = B, + B2 with BI self-adjoint and B2 skewsymmetric. Since B0 commutes with the orthogonal subgroup 60, so does Bo , as A* = A` for A E ®0. Hence, both B, and B2 commute with ®0. Since B, is self-adjoint, by Schur's lemma B, = cf for some
CER'. Since B2 is skew-symmetric, its minimal polynomial is the product
of k > 1 distinct irreducible polynomials, say PI, ... , pk. Furthermore, since it commutes with ®0, ker p;(B2) is an 00 invariant subspace of 60, 1 < i < k. Thus, ker pi(B2) is either (0) or Rd. Therefore,
k = 1 and p,(x) = x or p,(x) = x 2 + y for some y > 0. Note that a skew-symmetric operator has purely imaginary eigenvalues. Consequently, B. = cf + B21 where either B2 = 0 or Bz = - yI. Finally, we conclude that B = cf + W- I KBW, where KB is skew-symmetric and either KB = 0 or KB = - yl for some y > 0. From Theorem 4.6.5, we Q.E.D. obtain c z 1/2. Corollary 4.8.7. able, then B = cf.
Additionally, if either d is odd or B is diagonaliz-
Proof. Suppose d is odd. We know B2 is skew-symmetric. Then
det(B2) = 0. Hence, det(BZ) = 0, so B2 cannot be -yl for some y > 0. Therefore, B2 = 0 and B = cf. Now, suppose B is diagonalizable. Let A and v be an eigenvalue and corresponding eigenvector of B, respectively. Then, by Theorem 4.8.6, WKBW-'v = (B - cl)v = (A - c)v. Hence, A - c is an eigenvalue of KB. Since c and A are real numbers and since the real part of any eigenvalue of a skew-symmetric operator is equal to zero, c = A for every eigenvalue A of B, that is, B = cf. Q.E.D.
OPERATOR-STABLE MEASURES
236
4.9 THE CENTERING FUNCTION b(t)
In the characterization of a full operator-stable measure µ given in Theorem 4.2.12, that is, µ` = t8µ * S(b(t)) for all t > 0, the centering function b: (0, -) - Rd is unspecified. Its form can be described if 1 is not an eigenvalue of B. In this case, µ may be centered to eliminate b. This is analogous to the one-dimensional case since B = 1 gives the Cauchy distribution. Lemma 4.9.1. Let µ be t '-stable with centering function b: (0, oo) 9 d. Then, for all t, s > 0, b(st) = tb(s) + sBb(t)
(4.9.1)
and b(I) = 0.
Proof. For t, s > 0,
(s'µ * 3(b(s)))` = s A' * S(tb(s)) =,u" _ (st)Bµ * 3(b(st)) = sBµ` * S(b(st) - SBb(t)). Therefore, tb(s) = b(st) - sBb(t).Clearly, b(1) = 0.
Q.E.D.
Theorem 4.9.2. Let µ be tB-stable with centering function b: (0, co)
Rd and assume that 1 is not an eigenvalue of B. Then for some x0 E Rd we have, for all t > 0,
b(t) = (t1 - tB)xo.
(4.9.2)
Proof. When b(t) = 0 for all t > 0, select x0 = 0. Now, we only consider functions which nontrivially satisfy (4.9.1). Suppose c: (0, oo) Rd satisfies (4.9.1) and c(to) = 0 for some to * 1. Then toc(t0 1) _
c(t0-'t0) = t0c(t0-'), so to is an eigenvalue of to . But, 1 is not an eigenvalue of B, so to cannot be an eigenvalue of to . Hence, for any nontrivial solution c of (4.9.1), c(t) * 0 for all t # 1. Now, assume ca and c2 satisfy (4.9.1). Then c Cl - c2 also satisfies (4.9.1). Therefore, if two functions satisfying (4.9.1) agree for
some t # 1, then they agree for all t > 0. Clearly, the function
(t1 - tB)x, for fixed x E Rd, satisfies (4.9.1). Given the centering function b, let xl := b(2) and (21 - 28)x0 x1. Since 1 is not an eigenvalue of B, x0 is unique and always exists. Then b and the
CONTINUITY OF FULL OPERATOR-STABLE MEASURES
237
function (t - tB)xo satisfy (4.9.1) and agree at t = 2. Therefore, Q.E.D. b(t) = (t - t8)xo for all t > 0. Corollary 4.9.3. Let µ be as in Theorem 4.9.2 with centering S(-xo). Then v` = t By for all function given by (4.9.2). Let v
t>0.
The proof of this corollary is a simple calculation. Remark 4.9.4.
When 1 is an eigenvalue of B and µ is t B-stable, it
may not be possible to find an xo E Rd so that (µ * S(-xo))` = MA * 8(-x o)) for all t > 0. Consider the following example. Let d = 1, B = I, and µ(y) = exp(- IYI - i(2/ir)y loglyl) for y E U8'. Then µ is an asymmetric Cauchy distribution. For b(t)
(2/Tr)t log t
for t > 0, we have At = t'µ * 8(b(t)) for all t > 0, so µ is t'-stable with centering function b. It is easy to see that there does not exist an xo as in Corollary 4.9.3.
4.10 ABSOLUTE CONTINUITY OF FULL OPERATOR-STABLE MEASURES
In Theorem 3.8.9, it was shown that each full exp( - tQ)-decomposable
measure is absolutely continuous. It follows that each full operatorstable measure is also absolutely continuous. However, the proof in Section 3.8 was quite complicated, whereas a simpler proof for operator-stable measures is possible. This is done in this section. Moreover, it will be shown that, for each positive s and t, the functions IIYIISIi(Y)I` are Lebesgue integrable on 081. Thus, the probability density functions
of At, t > 0, are bounded and have partial derivatives of all orders [cf. Lukacs (1960)].
Lemma 4.10.1.
Let µ be full t'-stable on R' and let ip(y) :=
-loglµ(y)l for y E Rd. Then there are positive a, b, and c such that for, I11 a, +Ii(Y) ? CHYIIb
Proof. Since µ is infinitely divisible, Iii is finite everywhere. Also, µ being tB-stable implies that tpi(y) = *r(tB`y) for all t > 0 and y E Rd.
OPERATOR-STABLE MEASURES
238
By Corollary 4.2.14, ii(y) > 0 for all y E Rd \ (0). Let S:= (u E Rd: IIuIIB = 1) and c := min{r!i(u): u (=- S), where II ' IIB is the norm defined in Section 3.4. Since ,, is continuous and µ is full, we have c > 0. By Proposition 3.4.3, for each y # 0 there are unique ty > 0 and uy E S such that y = t B#u y. If IIYIIB ? 1, then t y >_ 1, so that IIYIIB 5 toBllR, and hence,
dr(y) = t/i(tB*uy) = tyi/i(uy) >_ cty >_ cIIYIIa
uBuB
Since all norms on Rd are equivalent, the above inequality implies the desired conclusion. Q.E.D. Theorem 4.10.2.
Let µ be full tB-stable on Rd. Then
(a) for all positive s and t, f dIIYllslii(Y)I` dy < oo;
(b) for each t > 0, the measure µ` has a bounded probability density function p, which has partial derivatives of all orders;
(c) for all t > 0 and all x e Rd, we have
pr(x) = where
t-$pi(t-B(x
- b(t))),
is the centering function for
trace of B.
and 13
Proof
(a) Let a, b, and c be as given in Lemma 4.10.1 and set A
[y E
Rd: IIYII > a). Then we have f IIYIIS(+G(Y))-t dy s c-` o
f
IIYII-«b -s>
dy
e
for all positive s and t. This last integral is finite for t sufficiently large. Therefore, for such t,
'if f r" ,
F(t) fd e 1(0(Y))
dY W J
239
DOMAINS OF NORMAL ATTRACTION
where I'(t) is Euler's gamma function. By the Fubini theorem, for a.a. r > 0, f IlYllse-"fr(y) dy
0. Clearly,
I
IlYllse-`0cy) dy < o0
R d\A
(b) For each t > 0, A' has a bounded probability density function p, since its characteristic function is integrable. Also, p, has
partial derivatives of all orders since all "moments" of its characteristic function are finite. (c) Simply calculate as follows: x
e-i<X,y>A'(y) dy
2ir fRd
Pr( )
fR e-i(x-b(t),y>%y(tB y) dy 2
t-
d
-i(t-B(x-b(t)),y>^
µ(Y) dy
27r fade
=
t-sp1(t-a(x
- b(t))).
Q.E.D.
4.11 DOMAINS OF NORMAL ATTRACTION OF OPERATOR-STABLE MEASURES
In the definition of an operator-stable measure p., it is assumed that (4.11.1)
with (A,,; E sue and v E 9. The obvious problem is, given µ, identify those measures v for which (4.11.1) holds. In this generality, we have the generalized domain of attraction, that is, v E GDOA(µ) provided (4.11.1) holds for some sequence ((A,,; a,,)) in W. This definition places no restriction on the form of A. Of special interest,
OPERATOR-STABLE MEASURES
240
suggested by the univariate case, is when A = n-B for some B E d(µ). This gives the domain of normal attraction, that is, v E DONA(µ) provided, for some sequence {an} in Rd, n-BV's * 8(an)
µ.
(4.11.2)
Hence, DONA(µ) c GDOA(µ), and the containment is proper. Also,
DONA(µ) # 0 when µ is full since µ itself is in DONA(µ) (cf. Theorem 4.2.12).
We use the following notation (cf. Theorem 4.6.5). Let µ be full and operator-stable on Rd with B E (µ). Decompose µ = µ, * µ2 and Rd = V, ® V2, where µ, is Gaussian with supp µ, = V, and µ2 has no Gaussian component with lin(supp µ2) = V2. Note that V, and V2 need not be orthogonal. Let P;, i = 1, 2, be the idempotents on V,.
Then P;(08d)=V,i=1,2,and P,P2=0=P2P,. Proposition 4.11.1. The exponent B commutes with each Pi, i = 1, 2. Thus, so does t B, t > 0.
Proof. For each x E Rd, Bx = BP,x + BP2x and Bx = P,Bx + P2Bx. Since V, and V2 are B-invariant subspaces, BP; x = P; Bx, i = 1, 2.
Q.E.D.
Lemma 4.11.2. Let µ be an operator-stable measure as described above. If p is an infinitely divisible measure on D such that Pip = µ;,
i= 1,2, then p=tt.
Proof. Let µ = [a, R, MI and p = [b, S, N]. Recall that A p = [b*, ASA*, AN] for A E Aut, for some b* E Rd (cf. Section 3.4). Since P2p = µ21 we have
P2SP2 = 0 and
P2N = M.
From PI p = µ we obtain
P,SPi =R and P,N = 0. Hence, supp N c V2, so N = P2N = M. Let S112 denote the self-adjoint square root of S. Then 0 = P2SP2 (P2S1/2XP2S1/2)*, so P2S112 = 0. Hence, P2S = 0 = SP2*. Consequently, S = (P, + P2)S(P1 + P2)* = P1SP; = R.
DOMAINS OF NORMAL ATTRACTION
241
From the uniqueness of the representation µ = [a, R, M], it now Q.E.D. follows that a = b. Therefore, p = µ. We now restrict our attention to the domains of normal attraction. We find necessary and sufficient conditions for v (=- .9(P) to be in DONA(µ), that is, v satisfies (4.11.2). We also show that for two different full operator-stable distributions either their domains of normal attraction are disjoint or the distributions are of the same type, in which case their domains of normal attraction coincide. It would appear that the DONA(µ) might depend on the choice of the exponent B of A. This is not the case. Proposition 4.11.3. Assume that v satisfies (4.11.2) for a particular exponent B of A. Then v satisfies (4.11.2) for every exponent of A.
Proof. Assume v satisfies (4.11.2) for some B E 6s(µ) and a,, E !Bd
and let B' E 6s(µ). Since both B and B' are in 6' (µ), we have nOµ * S(bn) = n = nBp * S(b',) for every n > 1, for some b,,, b', E QBd. Hence, with c,, n-B(bn - b;,), we have a,, :_ (n_BnB-; cn) E Inv(µ)
for all n >- 1a',(n_B-vn) (cf. Section 2.2.5). Also, (n-BnB')(nS(an) that is, µ, where an (n-BnB'; ar ). Since an' E
an'(a;,(n_BVn)) Inv(µ), we have, from Theorem 2.2.7, µ, that is, n-B' (nBa,, - b,, + b,,). Therefore, v n-B'vn * S(a',,) - µ, where a', satisfies (4.11.2) substituting B' and an for B and an, respectively.
Q.E.D.
Since µ can be decomposed into its Gaussian component µ, on VI and its Poisson component p-2 on V2, it is convenient to consider the
projections of v onto V, and V2. For v to be in DONA(µ), the projections PIv and P2v must behave properly. Theorem 4.11.4.
We have v r= DONA(µI * µ2) if and only if Pip E
DONA(µ;) for i = 1, 2. Proof. Assume v E DONA(µ), that is, (4.11.2) holds. By Theorem 4.6.5, with B; := BR', µ; is t B--stable on V for i = 1, 2. By Proposition 4.11.1, n-B' commutes with P,. Therefore, nB'(P,v)n * S(P,an) _ P,(n-BVn * S(an)) - P,µ, that is, P,v E DONA(µ,) for i = 1, 2. Now, assume P, v E DO NA(µ) for i = 1, 2. Then
n-Bj(Piv)n * s(bin) - µr for some sequences (b1n) and {b2n) in VI and V2, respectively, where
OPERATOR-STABLE MEASURES
242
BI Y and B E (p ). Consequently, P;(n -BV" * &(a )) - µ; for -BV
B;
i = 1, 2, where an _= b1n + b2,,. Hence, the sequence (n is
*
tight. Let p r= .9'(R") be the limit of some subsequence of
(n -BV" * S(an)). Then p is infinitely divisible and Pip = µ; for i = 1, 2. Q.E.D. By Lemma 4.11.3, p = µ. Therefore, n-BV" * t (an) = A.
To state our necessary and sufficient condition for v E DONA(µ), recall that if µ = [a, R, M] is full and operator-stable with an exponent B, then, for every E E .4(I8d \ {0}),
M(E) = f f IE(tBx)t-2 dtm(dx),
(4.11.3)
S. 0
where SV = (x (=- W': IIxII,, = 1) and m is a finite measure on . 9(Sµ) with supp m c V2 (cf. Propositions 4.7.9 and 4.3.4, Remark 4.7.12, and Theorem 4.3.7). Remark. In what follows, one could substitute the norm II the measure mB, with a fixed B E (f,(u).
- 11B and
From the central limit theorem (cf. Proposition 1.8.17), in order for v E DONA(µ) it is necessary and sufficient that for some B E f,(µ) we have
n n -BV(G) -> M(G) for every Borel set G which
is
bounded away from the origin and M(3G) = 0, (4.11.4)
and, for each y e Rd, lin-m-inf
Elm
limsup n
IIXII<E((Y,x))2(n
Bv)(dx) 2
_ ('11X11<E(Y,
x)(n-BV)(dx))
= (RY, y). (4.11.5)
From Theorem 4.11.4, we see that we may separately consider the cases when µ is Gaussian and when µ has no Gaussian component. We consider the "no Gaussian component" case first.
DOMAINS OF NORMAL ATTRACTION
243
Let µ = [a, 0, M] be full and operator-stable on In order for v E DONA(µ), it is necessary and sufficient that, for every F E 9(Sµ) with rn(a'F) = 0, Theorem 4.11.5. R".
lim t v((sBx: x E F, s > t)) = m(F),
too
(4.11.6)
where a'F denotes the boundary of F relative to Sµ, for some B E 4s(µ); in which case, (4.11.6) holds for all exponents of A.
Proof. We use the notation
[F; K] := {tBx:xEF,tEK}, where F E 9'(Sµ), K E 9((0, oo)), and B E d,' X0 is fixed. We know that, for each s > 0, sB[F; K] = [F; sKJ and W(Rd \ (0)) is generated by the family of all [F; K] (cf. Proposition 3.4.3). Note that a[F; K] = [cYF; K] U [F; a"K], where a"K is the boundary of K in (0, co). In particular, if the Lebesgue measure of lY'K is zero, then M(a[F; K]) = 0 if and only if rh(iF) = 0; simply note that [cf. (4.11.3)] for sets of the form [F; KJ we have
M([F; K]) = rn(F) fxt-2 dt. Now, assume that v E DONA(µ). Let F E M(S ) with th(a'F) = 0 and let K be a subinterval of (0, 00). Then M(3(F; K]) = 0 by the above. Consequently, from (4.11.3) and (4.11.4), we have for any B E 4(0 (cf. Proposition 4.11.3) n
v([F; nKJ) = n n-BV([F; K]) -, M([F; K]) = m(F) fKt-2 dt.
Hence, n P([ F; (n, cc)]) --> th(F). Letting [t] denote the greatest integer number, we see that
in(F) = lim [t] v([F;([t],oo)]) t -+ >_ limsup t v([F; (t,oo)]) >_ liminf t v([F; (t,oo)]) t- 00
t-+00
>_ lim ([t] + 1) v([F; ([t] + 1,00)]) = m(F). Therefore, (4.11.6) holds for any B E e,(µ).
OPERATOR-STABLE MEASURES
244
Now, assume (4.11.6) holds for a particular B E 6' (µ). Let F C 9(S,,) with m(a'F) = 0. Then, for s > 0,
n . (n-BV)([F;(s,°°)]) =n v([F;(ns,-o)]) t_2
s-'m(F) = t(F) f 00
dt.
S
Hence, for all subintervals K of (0, oo) we have
n n-BV([F; K]) -* h(F) fKt-2 dt = M([F; K]). Therefore, (4.11.4) holds because the sets [F; K] form a convergence class. We next show that lim lim sup n f E JO
IIXII2(n-BV)(dx) = 0,
(4.11.7)
IIxII'<E
n --00
from which (4.11.5) with R = 0 clearly follows. Let A
(Re(z): z is an eigenvalue of B).
Since µ has no Gaussian component, min A > 1/2. Let a and /3 be such that 1/2 < a < min A < max A _ e)) = t}) = m(F),
t- 00
then v E DONA(µ) and (a) and (b) hold for all exponents of A. Conversely, if v E DONA(µ), then (a) and (b) hold for all exponents
of µ. The final question about domains of normal attraction is: are they disjoint or do they overlap in some manner? Theorem 4.11.8. Let µ and y be full and operator-stable on R'. Then either DONA(µ) = DONA(y) or they are disjoint. Furthermore, DONA(p.) = DONA(y) if and only if the following three statements hold.
(a) µ and y are of the same type, that is, for some (A; a) E Qfl,
µ =Ay*S(a). (b) For any B E sequence (( n -BAn B;
there is a sequence in Rd such that the is relative compact in , where A is
from (a). (c) Every limit (D, v) of the sequence ((n -BAn B; vn)) in (b) belongs to Inv(µ), that is, p. = DA * 6(v).
Proof. First, we show that DONA(µ) n DONA(y) # 0 implies (a), (b), and (c). Let v E DONA(y) n DONA(y), B E d' (A), C E-= and (cs) sequences in fd such that d'(y), and and
y.
(4.11.10)
S(b - n-Bncc,,) - A. This convergence, (4.11.10), and Theorem 2.2.10 show that the set ((n-Bnc; b n-Bnccn)) is relative compact in Qfl and every limit (A; a) of this set satisfies µ = Ay * S(a). Hence, (a) holds. Then n-Bnc(n-cv" *
OPERATOR-STABLE MEASURES
248
From Proposition 4.6.6(b), A -'BA E 4(y), so for some sequence (c,',) we have n-A BAv" * S(cn Consequently,
Y.
(n-BAnB)(n-BVn * 6(bn)) * B(Acn + a - n-BAnBbn)
= n-BAv" * B(Acn + a) =
A(n-A-'BAVn
* S(c' )) * S(a)
- Ay * B(a) =A. This convergence with (4.11.10) and Theorem 2.2.7 shows that the set Ac' + a - n -BAnBbn, is relative compact ((n -BAnB; vn) ), with vn
in Vj and every limit of this set is in Inv(µ). Therefore, (b) and (c) hold.
Second, assume (a), (b), and (c) hold. Let v E DONA(µ). Then {bn}. From (b), we see that the sequence (n-BAnB(n-Bv" * B(b, )) * B(vn)) is tight. From (c), every convergent subsequence of this sequence converges to A. Conseµ. Hence, n-A BAVn * quently, n-Av" * S(n-BAnBbn + vn) n-BV" * B(bn) - µ for some sequence
8(A-'n-BAnBbn + A-'v,, - A-'a) - A-'µ * B(-A-'a) = Y. From (a) and Proposition 4.6.6, A-'BA E 6,(y), so v e DONA(y). Therefore, DONA(y) c DONA(y). The reverse inclusion is obtained in a similar manner, but there is a
slight difference. Let v E DONA(y). Then
n-A-'BAv"
for some sequence (dn). Hence, n-BA Vn * 5(Ad,, + a) A. Consequently, n-BAnBbn-BVn *
3(n-BA-'n8(Adn
* S(dn)
y
Ay * B(a) _
+ a - vn))l * B(v,,) = µ
Since every limit of the set ((n-BAnB; v,,)) is in Inv(µ), which is a compact group (cf. Theorem 2.2.5), this implies that nBvn * B(n-BA-'nB(Adn + a - v,,)) that is, v E DONA(µ). Therefore, DONA(y) c DONA(µ). Q.E.D. Remark 4.11.9. Theorem 4.11.4 suggests that any v E DONA(y) might be decomposed into v = VI * V2 with v, supported by V,.,
i = 1, 2. However, this is false. Consider the following example. Let d = 2 and µ((x, y)) := exp( _X2 /2 - I y D. Then µ is full operator-stable with B diag(1/2, 1) E d' (µ) with b(t) = 0 for all t > 0.
Note that Vl is the x-axis, V2 is the y-axis, A, is the standard
249
MOMENTS
univariate normal distribution, and 42 is the symmetric univariate Cauchy distribution. Let A, be any distribution on 982 which is concentrated on V, such that P,A, is a symmetric univariate distribu-
tion with a variance of 2. Also, let A2 be the distribution on 982, concentrated on V2, given by 4 ift< --,
2
7rItl
A2({(0, y): y 5 t})
1
2
4 4 if-- 0 be small and, by tightness, select a so large that a] < E for all n. Using Levy's inequality for symmetric summands, we have a])m
(P[IIAmnSnll 5
= P max11Amn(Snk - Sn(k-1))II < a] k<m
>_ 1 - 2P[IIAmnSmnll > a] > 1 - s
251
MOMENTS
for every n and m. Hence, P[IIAmnSnIl > a] < 1
For E E (0, 1), the quantity m(1 - (1 - E)'/n`) is bounded for all m >- 1. Therefore, sup mP[IIA,,,,Snll > a] < oo.
(4.12.3)
n, m
all A,,An,nll], using (4.12.2) and
Since P[II A,nnSnll > a] >(4.12.3), we have
acm'/s] < 00.
sup sup
n2N, mz1
Hence, there is to such that
t'/P] < oo.
N2 :, sup sup n2 N, t;,-, to
This implies that 00
sup E(IIAnSnIIP) = sup f P[IIA,,SnII z t'/P] dt nZN, 0
nZN,
- 1, the distribution of Un and the distribution of Vn is Anvn * S(an). Then, by (4.12.1), D
sup E(IIU, - V/IIP) < oo n
Let gn(y) - E(IIU,, - Y1') Then by the independence of Un and Vn we have Egn(Vn) = E(IIU, - V.11") 5 D.
Let b be so large that P[IIUnII > b] < 1/2 for all n. Also, let Bb (x e l : II x II -< b) and Gn (y E ll : gn(y) < 3D). We see that B.
OPERATOR-STABLE MEASURES
252
is not contained in W1 \ G, because if it were we would have D > E(gn(vn)IB6(Vn)) ?
3D 2
which is impossible. Hence, Bb n G # 0. Let yn E Bb n G for n>_ 1,
n-a 1 . Then, f o r
E(IIU -
E(IIUnII")
IIYnII)P
< CP(gn(Yn) + IIYn1IP) -< cp(3D + bP) < oo,
where cP is 1 when p < 1 and is 2P-' when p > 1. Therefore, sup,, E(IIAnSn + Hence,
oo.
oo for each n. Since IIAnSnII z IISnII/IIAn'II,
we have E(IISnIIP) < oo, so f II xII Pv(dx) < oo.
Since µ E GDOA(µ), we also have f II xII'A(dx) < oo.
Finally, by the uniform integrability, we have the convergence of moments as stated.
Q.E.D.
Corollary 4.12.3. If v E GDOA(.i1) and 0 < p < 2, then the conclusion of Theorem 4.12.2 holds. If v E DONA(.), then the covariance operator of v exists and is equal to the covariance operator of Xd; in particular, f IIxII2v(dx) < oo.
Proof. For the first statement, A = A = 1/2; and for the second statement, apply Theorem 4.11.6. Theorem 4.12.4. Let v c- GDOA(µ) with µ M(Rd) > 0. If p > 1/A, then
Q.E.D.
[a, R, M]. Assume
f IIxIIPV(dx) = 00.
Proof. Let C =_ (x e Rd: IIXIIB z a), where B is an exponent of µ [cf. (3.4.3)] and a > 0 is selected so that M(C`) > 0 and M(aC) = 0
253
MOMENTS
(cf. Corollary 4.3.6). Then f II xII Bv(dx)
f II An'xII B(Anv)(dr) C
> aPIIAnlIBP(Anv)(C).
Since n(Anv)(C) --> M(C) > 0, it suffices to show that n-'IIAnIIBP --' 5 cn-E for some oo. Let E E (1/p, A). By Theorem 4.5.3(a), constant c. Therefore, n-'IIAn1IBP >_ c-PnPE-' -* oo. Since the concluQ.E.D. sion holds for the norm II - IIB, it holds for any norm. Theorem 4.12.5. M(Rd) > 0. Then
Let v E DONA(p) with µ
[a, R, M]. Assume
(a) for p > 1/A, IlIx11Pv(dx) = 00; (b) for p < 1/A, JIIx11Pv(dx) < oo.
Proof. Recall the primary decomposition of l ' given by the linear operator B (cf. Theorem 4.6.5 and its proof). Let P be the canonical projection of Rd onto the subspaces corresponding to those eigenvalues of B having real parts equal to A. Then P is a polynomial in B, so P commutes with B. Since v E DONA(µ), we have n-BVn * S(an) - j u. Hence,
n-PB(Pv)n * S(Pan) = P(n
Bvn
* 3(an))
Pµ.
By Theorem 4.6.5, Pµ is full and operator-stable on PR" with exponent PB. Note that every eigenvalue of PB has a real part equal to A. Apply Theorem 4.12.5 to Pv and Pµ to obtain f II xII PV(dx) >_ f JJxIV '(Pv)(dx) = 00.
Q.E.D.
Summarizing these results for 0, we have the following theorem. Theorem 4.12.6.
Let
[a, R, M] be full and operator-stable.
Then
(a) M(18d) = 0 implies JIIxlI' (dx) < c for all p > 0; (b) M(Rd) > 0 implies
(i) JIIxIIPµ(dx)
is
given
by A(x)
Proof. By adding and subtracting Yj, Jjx )
1 + Ej_1CJjx, Jjx)
in the appropriate place in the characteristic function of v, this is an Q.E.D. easy calculation.
The following result on independent marginals applies to any infinitely divisible measure, not necessarily operator-stable. Theorem 4.13.2. Let µ = [a, R, M] be full and let J1µ, ... , J,, 1L be marginals. Then J1µ, . . . , Jrµ are independent if and only if for all j : k we have both
JJRJk = 0,
(4.13.1)
M((ker Jj)` n (ker Jk)`) = 0.
(4.13.2)
Proof. Since J1µ, ... , Jrµ are independent if and only if their joint characteristic function is the product of their individual characteristic functions, we see from Lemma 4.13.1 that they are independent if and only if, for all y1, ... , Yr E Rd r
E `JjRJJ yk, Yj) =
j, k-1
L.r \ JiRJi*Yi, Yi),
yr; X1,..., r
j=1
f
Rd\(p)91(yj;
(4.13.3)
i-1
xj)(JiM)(dxj).
xr)(AM)(dxl,...,
dxr) (4.13.4)
OPERATOR-STABLE MEASURES
256
Obviously, (4.13.1) and (4.13.3) are equivalent. We wish to rewrite X 8(0) and for B1,..., Br E , (Rd) with Bi x (4.13.4). Let So \ (0)) as a product Br E R(l8rd \ (0)) define the measure Xfl on measure by q(Rrd
xBr)
M(Bi x . r
SofB i) ... SofB;-i}( JiM)(B;)Sa(Bi+i) .
. .
So(Br)
i=i
Then, for an M-integrable function f : R rd -3, U8', we have
f (Xi, ... ) Xr)M(dXi,... , dXr) RrdS{0}
r
Ef
f(0,...,O,X1,O,...,0)(JJM)(dxi)
J=i Rd-\10)
Hence, M is a Levy measure and (4.13.4) is equivalent to
(AM)(B1 x ... X Br) = fv0(Bi X ... X Br).
(4.13.5)
Therefore, it suffices to show that (4.13.2) and (4.13.5) are equivalent. Assume (4.13.5) holds. For fixed j # k, let BB = Bk = Rd \ (0) x Br) = 0 and x B,) = M((ker J7 n (ker Jk)`), so (4.13.2) holds. Now, assume (4.13.2) holds. If there are j 0 k with 0 t4 B1 and xBr)< xBr)=0 and (AM)(B1x 0iZ Bk, then M(B1x M((ker JJ)` n (ker Jk)`) = 0, so (4.13.5) holds in this case. If 0 B' for some j, but 0 E B, for all i # j, then J7 '(B,) C (ker J1)`, Ji-'(B,) c (ker J;)` for i # j, and
and let the other B,'s be Rd. Then M(B1 x
(AM)(B1 x
(AM)(B1 x ... X Br) = M( n (J; '(Rd) i#J = M(JJ
\J,-'(B, )))
(nJi-'(R
d)))
'(BJ) n = (J;M)(Bi) = M(Bi x ... x Br), where the first equality uses (4.13.2), so (4.13.5) holds in this case. (A slight change in the argument is needed when r = 2.) Since 0 cannot Q.E.D. be in every B;, we have (4.13.2) implies (4.13.5).
257
INDEPENDENT MARGINALS
This theorem shows that for marginals of an infinitely divisible law
to be independent it is necessary and sufficient that pairs behave properly.
Corollary 4.13.3. Let p. be full and infinitely divisible on Rd. The marginals J1µ, ... , JA are independent if and only if they are pairwise independent.
Remark 4.13.4. It is not necessary for the marginals to be on orthogonal subspaces in order for them to be independent, because (4.12.2) may hold without the subspaces being orthogonal subspaces.
Lemma 4.13.5. Let µ = [ a, R, M ] and let J1µ,
... , Jrµ be
marginals. Then the condition (4.13.2) implies r
supp M c U ker J,.
(4.13.6)
i=1
When J1µ, .... Jrµ are a complete set of orthogonal marginals, then (4.13.2) is equivalent to r
supp M c U JJ(V).
(4.13.7)
I=1
Proof. If (supp M) n (U j=1 ker JJ)`
0, then (supp M) n
(ker J)` n (ker J2)c # 0, which contradicts
(4.13.2). Therefore, (4.13.2) implies (4.13.6). When we have a complete set of orthogonal marginals, I = E j= 1 J, and Jj Jk = 0 = Jk J; for all j * k. This implies that r
U Ji(V) = n ((kerJj) U (kerJk)). J=1
)#k
Clearly, this equality shows that (4.13.2) and (4.13.7) are equivalent. Q.E.D. One might wonder whether there are different choices for independent univariate marginals, possibly only one choice if we also require them to be complete and orthogonal. However, if µ is Gaussian and full on Rd with d ? 2, there are obviously an infinite number of such univariate marginals, since marginals of a Gaussian are Gaussians and
OPERATOR-STABLE MEASURES
258
they are independent when on orthogonal subspaces. But, if µ has a "significant" Poisson component, we see in the next theorem that µ has at most one choice for such marginals. Proposition 4.13.6. Let µ = [ a, R, M ] and assume [ a, 0, M ] is full. If µ has a set of complete independent orthogonal univariate marginals, then µ has exactly one such set of marginals. Proof. Suppose J11L, ... , Jdµ and K1,..., Kdµ are sets of complete independent orthogonal univariate marginals. By Theorem 4.13.2 and Remark 4.13.5, d
d
supp M C ( U i
n U Ki(EJBd)
1
.
i=1
Fix i. Since [a,0, M] is full, lin(supp M) = 1W' (cf. Chapter 1). Hence, d
U suppo m n J;(QRd) n K, (Rd)) = suppo M n J,(Rd) * 0, j=1
where suppo M (supp M) \ (0). Hence, there is j(i) such that suppo M n J,(Rd) n KJ(JRd) # 0. Since J;µ and KJ(;)µ are univariate, we have Jj(1W') = Kj(,)(1W'). By orthogonality,
JkKf(;) = 0 = KJ(j)Jk,
i # k.
Using completeness, d
J,
_ E J,Ki(k) = Ji K;(1) k=1 d
E JkKi(j) = K;cj>
Q.E.D.
k=1
We now specialize to full t '-stable measures µ = [a, R, M]. The decomposition of such µ given in Theorem 4.6.5 and the information about B given in Theorem 4.6.12 will be used in stating and obtaining the results. To fix the notation in the remainder of this section,
A = Al * A2 and Rd = V 1 ®V2 ,
259
INDEPENDENT MARGINALS
µ, is full Gaussian on V,, µ2 has no Gaussian component and is full on V2, and the subspaces V, and V2 correspond to those eigenvalues of B with real parts equal to 1/2 and greater than 1/2, respectively. Let L, and L2 be the projections of pd onto V, and V2, respectively, Note with L,L2 = 0 = L2L,. Then L;µ = µ;. Also, µ; is that V, and V2 need not be orthogonal. The following is a main result tLiB-stable.
concerning operator-stable measures.
Theorem 4.13.7. Let µ be full and operator-stable on Rd. If dim V, S 1 and if J1, ... , Jdµ and K,,u, ... , Kdµ are two sets of complete orthogonal independent univariate marginals, then {J1,. .. , Jd}
= (K,,..., Kd). Proof. When dim V, < 1, µ = [a, 0, M], so apply Proposition 4.13.6.
Now, assume dim V, = 1. By Lemma 4.13.5, supp M c U d 1Ji(Rd) Hence, J;(Rd) C V2 for i such that suppo M fl J;(Rd) 0. Since J, µ, ... , Jdµ is a set of complete orthogonal univariate marginals, there is a unique j such that V, = JJ(Rd) and V, n JJ(Rd) = (0) for i j. Hence, we may assume J, = L, = K,. For i > 1, J;µ = J,µ2 and
K;µ = K;92. Considering µ21 J and K. for 2< i < d on V2, by Proposition 4.13.6 we have {J2, ... , Jd} = (K2,..., Kd) . 1
Q.E.D.
Remark 4.13.8. When dim V, > 1, Theorem 4.13.7 is false, because the projections into V, are not unique, whereas the projections into V2 are still unique.
For µ = [a, R, M] E .9Th 05 with univariate marginals J,µ, , Jdµ, let T,
T2 := {j: dim JJ(V2) = 1).
(j: dim JJ(V,) = 1),
Theorem 4.13.9.
Let µ = [a, R, MI be full and operator-stable on
Rd and let J,µ, ... , Jdµ be a set of complete orthogonal univariate marginals. Then they are independent if and only if
R = F, JJRJ$*,
(4.13.8)
j e T,
supp M C U Jpd).
(4.13.9)
jeT2
Proof. Let T :_ (j: suppo M fl JJ(Rd) 0). Assume the marginals are independent. They by Theorem 4.13.2 and Lemma 4.13.5 we
OPERATOR-STABLE MEASURES
260
have supp M c U"1JJ(Rd). Hence, supp M c U iETJJ(Rd). But, lin(supp M) = V2, so obviously T = T2; hence, (4.13.9) holds. Since µ2 has no Gaussian component, (L2R1/2)(L2R1/2)* = L2RL2 = 0, so L2R = 0. Hence, for j E T2, JJR = 0. By Theorem 4.13.2, JJRJk = 0 for j # k. Therefore, R = L, JJ RJ,! _ F. JJ RJJ*. j,k jET,
The converse part of this theorem follows by noting that (4.13.9) implies (4.13.6), and that (4.13.8) with the first equality above implies (4.13.1). Q.E.D. Theorem 4.13.10.
Let µ = [a, R, M] be full and operator stable on
Rd and let J1,. .. , Jdµ be a set of complete orthogonal independent univariate marginals. Then
(a) the marginals J1µ1, ... , Jdµ1 are independent; (b) the marginals J1µ2, .... Jd/ 2 are independent; (c) for j E Ti, Jig is univariate Gaussian; (d) for j E T2, Jig is univariate stable non-Gaussian;
(e) there are aJ > 1/2 for j E T2 such that the linear operator
2 L1 +
r`
jET2
a1JJ
is an exponent for A. kd2
Proof. As in Lemma 4.13.1, define A: Rd \ (0) (J1x,... , Jdx). By independence, for y1, ... , Yd E
N (0) by A(x)
IlBd
d
(AA)^(Y1,...,yd)= r1(JjA)^(Yj) j=1
Since µ = µ1 * µ2,
(AA)^(y1,...,Yd) =
(AIl1)^(y1,...,Yd)
- (Aµ2)^(y1,...,Yd).
For fixed j, set y; = 0 for i # j to obtain (Jig )^ (Yi) =
(J1µ1)^ (Yi) . (1Jµ2)^ (Yj)
261
MULTIVARIATE STABLE MEASURES
Hence, (YI, ... , Yd)
(AIAI )^ (Y1, ... , Yd) ' d
d
j=1
c=1
This proves (a) and (b) by the uniqueness of the decomposition of an infinitely divisible law into its Gaussian and non-Gaussian components. For (c) and (d), Jiµ = J;µ1 * Jµ2. Then Jiµ is Gaussian if and only
if Jiµ2 = 0. For j E T1, J1V2 = 0, so Jiµ2 = 0. Also, Jiµ has no Gaussian component if and only if Jiµ1 = 0. For j E T2, JJV1 = 0, so
Jill, = 0. Clearly, (e) follows from (c) and (d).
Q.E.D.
Corollary 4.13.11. Under the assumptions in Theorem 4.13.10, µ has exactly one exponent if and only if dim V1 5 1.
Proof. If dim V, > 1, the Gaussian measure µ1 on V1 has many exponents. This implies that µ also does. If dim V1 5 1, the uniqueness of the exponent of µ follows from the uniqueness of the characteristic index of any univariate stable measure.
Q.E.D.
Corollary 4.13.12. Let µ be full and operator-stable on Rd. Then µ has a set of complete orthogonal independent univariate marginals if and only if there is an exponent of u. which is diagonalizable.
4.14 MULTIVARIATE STABLE MEASURES
We say that µ E -60(R d) is multivariate stable if there are an > 0, u E Rd, and V E _6p(Rd) such that Tanv" *
In particular, when v = 0 in (4.14.1), we say that µ is
(4.14.1)
strictly
multivariate stable. For brevity, we use the terms stable and strictly stable. Recall that, for a e R', Tax ax for all x E Rd. Thus, stable measures are a special subclass of the operator-stable measures obtained by requiring the norming sequence (AR) to be of the restrictive with an > 0. Therefore, the results in the earlier form An =
262
OPERATOR-STABLE MEASURES
sections of this chapter can be specialized to this setting. However, the
assumption that µ be full can be weakened to assuming that p. is nondegenerate, that is, µ 0 S(a) for any a E R d. Also, some proofs are much simpler and straightforward than their counterparts in the general framework of operator-stable measures. Note that {Ta v: n >_ 1} is an infinitesimal array (cf. Section 4.1), and that when µ is nondegenerate, we have an -> 0 and an+1/an -> 1 as n -+ oo (cf.
Proposition 3.9.1).
Proposition 4.14.1. A measure µ E _AR d) is stable if and only if for every n >_ I there are bn > 0 and xn e 68d such that
µn = T,p. * S(xn).
(4.14.2)
In particular, µ is strictly stable if and only if xn = 0 in (4.14.2).
Proof. When µ is degenerate, (4.14.2) is obvious. Assume µ is nondegenerate and stable. Fix n z 1. Then Takvkn * S(nvk) - An as Taknvkn * S(vkn). Then µk µ and Takakn'Ak k - oo. Let µk S(nvk - akakn vkn) - µn as k - oo. By Proposition 2.2.11, there are b n > 0 and xn E i8d such that (4.14.2) holds. The remainder of the proof is obvious.
Q.E.D.
We now obtain a particular form of (4.2.1) when µ is stable, but not necessarily full. We cannot simply apply Theorem 4.2.12 because its proof used the symmetry group of operators which is neither a group nor compact when µ is not full. However, since for µ stable we are limited to multiplying by constants, the analogous symmetry group is a compact group.
Theorem 4.14.2. Let µ E . (flB') be nondegenerate. Then µ is stable if and only if there is a > 0 such that for all t > 0 there is
x, E Rd with µ` = 1 1/..p * S(x,)
(4.14.3)
Proof. When (4.14.3) holds, µ is stable by Proposition 4.14.1. Now
assume µ is nondegenerate and stable. For t > 0, let G, :_ {b # 0: A' = Tbµ * S(x) for some x (=- Rd} and let G == U 1 > 0G,. Using Proposition 4.14.1 and making slight changes in the proofs of Lemmas
4.2.3 to 4.2.7, we see that G is a closed subgroup of the group (W \ (0), - ) and that G1 is a compact subgroup of G. So, select c > 0
263
MULTIVARIATE STABLE MEASURES
such that e` 4t G1. Recall that the function 1: G - R8
given by t if b E Gr is a continuous homeomorphism (cf. Lemma 4.2.8). 1(b) log 1(es`) is a continuous homeoSince h: R' --> 118' given by h(s) morphism (cf. Lemma 4.2.9), there is a # 0 such that h(s) = as for all
s. Let a
a/c. Then log 1(es'") = s, so log 1(t'I") = log t for all
t > 0, that is, t'l" E G,. Therefore, (4.14.3) holds.
Q.E.D.
Using this theorem and the proof of Theorem 4.6.5, we obtain the following corollaries.
Corollary 4.14.3. Let µ E 9(R') be nondegenerate. Then µ is stable if and only if (4.14.3) holds for some a e (0, 2]. Furthermore,
when a = 2, µ is Gaussian, and when a < 2, a has no Gaussian component.
Corollary 4.14.4.
with 0
- 1 - it(X,
z
)
t-('+a)dtm(dx)
.
It is obvious that if µ is stable on Rd with exponent a, then, for each x E Rd, IIXµ is univariate stable with the same exponent a, where IIx(y) := (x, y). Is the converse true? That is, do all the univariate marginals of µ E p(Rd) being stable imply µ is stable on Rd? In this generality, the answer is no! We shall show that with the added assumption that µ is infinitely divisible the answer is yes. Lemma 4.14.7. Let µ E .9(R' ). If for all x E Rd the univariate distribution IIxµ is strictly stable, then each distribution in the set (IIXµ:
x E Rd and IIXµ # 3(0)) has the same exponent a E (0, 2].
Proof. Let a(x) denote the exponent of IIXµ with the convention
that a(x) = 2 when IIXµ = 8(0). For each a E (0, 21, let F(a) {x E Rd: a(x) >- a}. We first show that each F(a) is a subspace of Rd. For t # 0, IIXµ and IIXµ have the same exponent, because
(s) = µ(stx) = (IIXµ)^ (st).
Hence, tx E F(a) for all t and for all x E F(a). Now, let x, y e F(a) and let {4} be a sequence of independent identically distributed random vectors with common distribution A. Then, for each n >- 1, n-1/a(X)IIx( 1 + and 11x 1 have the same distribution, and n-'/a(y)IIy(61 + +0 and IIy61 also have the same distribution. Hence, for any e E (0, a), _Z(n-'/EIIx+y(61 + +0) - 3(0).
MULTIVARIATE STABLE MEASURES
265
Consequently, e < a(x + y) for every e E (0, a), so a < a(x + y). Therefore, x + y E F(a), and F(a) is a subspace. Second, let aF (a(x): x E F), where F is any subspace of lid. Then, if dim F = 2, aF cannot have more than two elements. To see this, suppose to the contrary that there is a subspace F of dimension 2
with x, y, z E F for which a(x) < a(y) < a(z). Then no two of x, y, z can be colinear since for colinear vectors w,, w2 we have a(w,) = a(w2). Since F has dimension 2, x = tl y + t2z for some nonzero t t2. Since F(a(y)) is a subspace and y, z E F(a(y)), we have x = tly + t2z E F(a(y)), that is, a(x) >_ a(y), a contradiction. Third, assume F is a two-dimensional subspace such that there are
all a2 E aF with a, < a2. Then aF = (a,, a2). Let F2 F rl F(a2). Then F2 is a subspace of F and dim F2 < 2 since a, < a2. For x E F2, select x,, E F \ F. such that x -> x. Then n.,.µ IIxµ (cf. Corollary 1.7.3), and the exponent of IIxµ is a2, whereas each IIx µ has exponent a,. This is not possible, unless a2 = 2 and IIxµ = 8(0). Finally, since any two points in Rd lie in some two-dimensional subspace, we obtain the lemma. Q.E.D. Theorem 4.14.8.
Let µ E 9( d). Then µ is strictly stable on Rd if
and only if IIxµ is strictly stable on R' for each x E Rd. In which case, µ and IIxµ have the same exponent a whenever flu = S(0). Proof. The converse part is immediate, so assume that each IIxµ is strictly stable. By Lemma 4.14.7, the exponent of each IIxµ is the
same a for each IIxµ # 3(0). When IIxµ = 3(0) for every x, the theorem is obvious. Hence, assume IIxµ
5(0) for at least one x. We
will show that given a > 0 and b > 0, there is c > 0 such that Taµ * Tbµ = Tcµ. For x with IIxµ # 3(0), select c > 0 so that Tallxµ * TbIIxµ = TcIIxµ. Since IIxµ is strictly stable with exponent
not depending on x, such c exists and does not depend on x. Set v := TaIcµ * TblCp. Since T and II commute, IIxv = H. so IIxv is strictly stable. Hence, v = Al, so. Tap * Tbµ = Tcµ. Now, by induction,
for every n z 1, µ" = Tbµ for some b > 0. By Proposition 4.14.1, µ is strictly stable.
Q.E.D.
Now, we have an obvious corollary. Corollary 4.14.9. Let µ E .I(R d). If each IIxµ is stable on 68', then µ° is strictly stable on 1 '.
266
OPERATOR-STABLE MEASURES
This brings us to a main result. Assume µ is infinitely divisible on Rd. Then µ is stable on LW' if and only if IIxµ is stable on R' for all x E Theorem 4.14.10.
Rd.
Proof. When µ is stable on Rd, it is obvious that each H,,µ is stable
on R. So, we now assume that each IIxµ is stable on R' with exponent a E (0, 2).
First, assume 1 < a < 2. Let m denote the mean of µ and let v
µ * S(-m). Then each IIxv is strictly stable on R'. By Theorem
4.14.8, v is strictly stable on IWd, so µ is stable on LW'.
Second, assume 0 < a < 1. Since IIxµ is stable, IIxµ° is strictly stable. Hence, µ° is strictly stable on Rd. Consequently, µ has no Gaussian component. Let p. = [a, 0, MI. Then M + M- is given by (4.14.4), where M-(A) := M(-A). Hence,
f
1
IixII M(dx) _ - f
Ilxll(M + M-)(dx)
2 1Ixu51
uxuS1
m(Sd-1)
2(1 -a)
- 1)N(dv)}
267
MULTIVARIATE STABLE MEASURES
for some Levy measure N (cf. Theorem 4.14.5). Hence, for all x, Ilxv * S((x, b)) = IIxµ = nxp * S(ax). By the uniqueness of the representation of an infinitely divisible measure, ax = (x, b) and v = p. Therefore, µ = p * 8(b), so µ is stable on Rd. Third, assume a = 1. As before, µ° is strictly stable with exponent 1. Hence, µ has no Gaussian component, so we can write, for some a E Rd, µ = v * S(a), with P(y)
exp{ f (e+ - 1 - i(Y,v)111
1Is11)M(dv)}.
(4.14.5)
For notational convenience, denote any measure v whose characteris-
tic function v is of this form by cl,d(M), for d-dimensional and centered at 1 with Levy measure M. Then, for all x E lied, there are ax E W' and c,, I(Mx) such that 11zµ = cl,,(Mx) * S(ax), with cl,,(Mx) being stable with exponent 1.
Since µ° is strictly stable with exponent 1, Hence, by Theorem 1.7.1(b), the sequence {T"-,µ": n 1} is shift tight. Therefore, letting m f(llxll s l,xµ(dx), the sequence (T"-,µ" * S(-m"): n > 1) is tight, and any measure 77 which is the limit of some subsequence of it is of the form n = cl,d(N), with N possibly depending on the subsequence. Select such as subsequence so that
T,,-'µ" * s(-mn) - ci,d(N)
(4.14.6)
along that subsequence.
Since IIxµ is stable with exponent 1, for each n >: I and each x E Rd, there is j(n, x) E R1 such that T"-,(11iµ)" * S(j(n, x)log n) = n,,µ. Hence,
T"-Olxµ)" *
5(m" + j(n, x)log n) = IIxµ. (4.14.7)
From (4.14.6) and (4.14.7), there is hx E l1' such that m" + An, x)log n --> hx along that subsequence. Consequently, from (4.14.5) and (4.14.6),
nxcl,d(N)*S(hx) = nxµ
= IIxcI d(M)*S((x,a)).
(4.14.8)
OPERATOR-STABLE MEASURES
268
For any Levy spectral measure G of a stable measure with exponent a = 1 and any x E Rd, let
c(G, x) := I
I[,i,;us1)(x, v)G(dv).
Note that c(G, x) exists by Theorem 4.14.5. Then
ilxcl,d(G) = c1.1(IlxG)* 5(c(G, x)). Hence, (4.14.8) becomes c1,1(IIxN) * 3(hx + c(N, x)) = C1 , 1(rlxM) * 5((x, a) + c(M, x)).
Thus, HxN = IIxM for all x. In particular, we have c(M, x) c(N, x) = (x, b), for some b E Rd. Consequently,
llxcl,d(M) = c1,1(11xN) * S(c(M, x)) = IIxc1 d(N)* 5(llxb).
Therefore, c1,d(M) = cl,d(N)* 5(b), so b = 0 and M = N. From (4.14.5), p. = c1 d(N)* 5(a) with c1,d(N) being stable. Therefore, µ is Q.E.D. stable on R' Example 4.14.11. It can be shown that the complex-valued function g defined on R 2 by
g(x, Y)
ely.cos3e
- r«,
where (r, 0) are the polar coordinates of (x, y), 0 < a < 1, and y > 0 is sufficiently small, is the characteristic function corresponding to a nonstable measure on 682. However, all its univariate marginals are stable with exponent a. What is surprising is that Example 4.14.11 essentially captures the necessity to assume that µ is infinitely divisible in Theorem 4.14.10. This assumption can be weakened. Theorem 4.14.12. Let µ E ,9(68') be such that all of its two-dimensional marginals are infinitely divisible. Then µ is stable on Rd if and only if each 1-I,µ is stable on 681.
269
MULTIVARIATE STABLE MEASURES
Proof. Assume each H.
is stable on R'. Let u1, u2 E Rd be
linearly independent, let V
lin(u,, u2}, and let J be the orthogonal
projection of Rd onto V. for x e R , let x'
Jx. Then IIx(Jµ) _
IIX.µ. Since each IIX.µ is stable on R' and Jµ is infinitely divisible by
assumption, by Theorem 4.14.10 we have that Jµ is stable on V. Therefore, given a > 0 and b > 0, there are c > 0 and f(ul, u2) E V such that TT,(JA) * Tbc(JA) * a(f(u1, u2))
= J(Tacµ * Tb,A) * s(f(uII u2)) = Jµ.
(4.14.9)
For x E Rd, let g(x) E R be such that IIX.(TQdµ * TbCµ) * s(g(x)) = IlX,µ.
(4.14.10)
Using IIX(Jµ) = IIX,µ, (4.14.9), and (4.14.10), we see that g is linear
0. Therefore, there is z such d. Also, if x - 0, then that g(x) = (x, z). Hcncc, IIX.(Ta p * Tbcµ * S(r)) = IIX µ, where r E Rd is selected so Jr = z. This implies that all univariate marginals
on 1
of TQcµ * Tbcµ * S(r) and of µ agree. By the Cramer-Wold device, the Q.E.D. two measures agree, so µ is stable. Example 4.14.13 (Levy). Let X and Y be independent real-valued
random variables, each .A"(0,1/4), and let µ be the distribution of the random vector (X2 , 2XY, Y 2). Then p. is not infinitely divisible, but the joint distribution of any two of X2 , 2XY, and Y 2 is infinitely divisible.
Example 4.14.14. There are noninfinitely divisible measures on Rd
with d > 2 which have all of their two-dimensional marginals being infinitely divisible. For y E Rd, d > 2, define P(y) := exp{ f (e'(r.X) - 1)ZE(dx)l,
where ZE(F) := ,1(F n D,) - eA(F n D2) for 0 < e < 1,
A
is a
Lebesgue measure on D", DI (x E Rd: 1 < IIxII < 2), and D2 (x E Rd: IIxll < S), with S > 0 small. Then p is a characteristic function for each e and 3. Also, by selecting 3 > 0 sufficiently small, the two-dimensional marginals of Z. are positive measures. Consequently, these marginals are infinitely divisible.
OPERATOR-STABLE MEASURES
270
Let us now examine the domain of attraction of a stable measure on 0 d. For a stable measure p. with exponent a E (0, 2], its domain of normal attraction, DONA, consists of those v c= .9`(Pd) such that n (t/a)IVn * l5(vn)
p.
for some sequence (vn} in Rd
Let µ be stable on Rd with exponent a. When a = 2, so µ is Gaussian, v E DONA(µ) if and only if the covariance operators of p. and of v are the same. When a < 2, v r= DONA(µ) if Theorem *4.14.15.
and only if
t'Ic)) = m(E)
lim
for every E E .(S d - '), where m is given in Theorem 4.14.5.
This theorem is a consequence of Theorem 4.14.5. Next, we consider the finiteness of moments. When µ is Gaussian,
the results are contained in Theorem 4.12.2, Corollary 4.12.3, and Proposition 4.12.4. For µ non-Gaussian, from Theorems 4.12.5 and 4.12.6 we obtain the following result. Theorem 4.14.16. Let µ be stable on Rd with exponent a < 2. Then J II x II Pp(dx) < oo if and only if p < a.
Finally, we consider the problem of the existence of a set of complete orthogonal independent univariate marginals (cf. Section 4.13). The Gaussian case is immediately obvious, so we address the non-Gaussian case. Theorem 4.14.17. Let µ be stable on Rd with exponent a < 2. Then there is exactly one set of complete orthogonal independent univariate marginals. Furthermore, Jl µ, ... , Jdµ is such a set if and only if the Levy measure M of µ satisfies d
supp M c U J1(Pd) 1=1
Proof. By Corollary 4.13.12, such a set of univariate marginals exists, and, by Theorem 4.13.6, there is only one such set. The final statement follows from Theorem 4.13.9. Q.E.D.
THE CASE WHEN d = 3
271
4.15 THE CASEWHENd=3 Let µ be full and operator-stable on 183. We know that its symmetry
group, A(µ), is conjugate to a closed subgroup, ®o, of the full orthogonal group ® on 183 (cf. Theorem 2.4.1). Lemma 4.15.1.
The dimension of the tangent space J (®o) is either
0, 1, or 3.
Proof. We know that 9-W) is X, the set of all skew-symmetric operators on I183 (cf. Example 1.5.7). Since O, is a closed subgroup in
®, we have that .l(®0) c .l(®) and that Y(00) is closed under the Lie bracket [, ] (cf. Lemma 1.5.3). Define the linear transform f from II83 onto .
by
-a -b 0 -c
0
f(a, b, c) := a b
c
.
0
[ f(x), f(y)] = f(x x y), where x is the usual cross product on 183. Hence, (183, x ) is isomorphic to X. Note that, for x, y E l Therefore, either
-'7(®o)
,
= {0}, dim J-(®0) = 1, or ®o = ®.
Q.E.D.
When Y(t9o) = {0}, µ has an unique exponent. When A(µ) is conjugate to ®, the class of exponents d(µ) is given by cI + W-'.'W for some c >- 1/2 and for some W E Aut(183) (cf. Theorem 4.8.1), and
the characteristic function µ, is given in Theorem 4.8.4. We now consider the case when dim .17(®o) = 1. Lemma 4.15.2.
Let ®o be a closed subgroup of ® with dim ®o = 1.
Then
5-(do)=
0 c 0
-c
0
0
0 :cE18' ,
0
0
with respect to some orthonormal basis for O.
Proof. We know dim Y(eo) = 1, so let {Q} be a basis for Y(®o).
Then Q E X. Since det Q = det(- Q*) = (-1)3 det Q, det Q = 0, that is, Q is singular. Let u E R 3 be such that Jul = 1 and Qu = 0. Let this u be a third member of an orthonormal basis for 183. This is
OPERATOR-STABLE MEASURES
272
the basis needed in the lemma. By skew-symmetry the third row and third column are all zeros, and by skew-symmetry the diagonal is all zeros and the (1, 2)-element is the negative of the (2,1)-element. Since Q.E.D. {Q} is a basis for 97(0o), we have the stated result. Theorem 4.15.3.
Let µ be full operator-stable on 1183. If 9-(A(µ)) is
neither (0) nor conjugate to X, then there is a positive-definite selfadjoint linear operator Won R3 and there are real numbers a, b Z 1/2 such that, with respect to some orthonormal basis for R', a c 0
-c
0
a 0
b
0 :cEl 1
Proof. Since A(µ) is conjugate to a closed subgroup ®o of
',
select the W so that A(W-'µ) = ®o. By Lemma 4.15.2, J(A(W- lµ)) =
0 c 0
-c
0
0
0: c E U8'
0
0
with respect to some orthonormal basis. Use this basis for all matrix representations. Let B E 9'.,(W-'µ) with the representation (b;j). By Lemma 1.5.3 and the proof of Theorem 4.6.7, BQ - QB E
T(A(W-'µ)) for all Q E T(A(W-'µ)), that is,
B(0
0)
- (0
0)B =
c 0
0 0
0 0
for some c e R1, where
J :_ This implies that b31 = b32 = b12 = b23 = 0 and b12 = b21. It only remains to show that b11 = b22. We now know that B' := diag(b11, b22, b33) is in 4(W-'µ). Since B' commutes with every Q in T(A(W-'µ)), we have that the diag(bll, b22) commutes with every 2 x 2 skew-symmetric matrix. Using Schur's lemma, this implies that diag(b,,, b22) is a multiple of the Q.E.D. identity. Hence, b11 = b22.
273
THE CASE WHEN d = 3
Theorem 4.15.4.
Assume dim(.l(A(µ))) = 1 and
--T(A(µ))=
0 c 0
-c 0 0
0 0 0
cE68'
µ has an exponent B whose matrix representation is diag(a, a, b) with respect to the usual basis, where a and b are both greater than 1/2, and M is the Levy measure of A. Then we have that A(µ) = 0', where 0' is the subgroup of 0 generated by all orthogonal transformations which leave the z-axis invariant. Furthermore, we have that there is a finite Borel measure v on [ -n/2, 7r/2] such that, for Borel A c 683 \ {0),
M(A) =
1-'/2 /2 0f 2, f0IA(tBx(0, (p))t-2 dtdO v(dcp),
where x(9, cp) = (cos 0 cos (p, sin 0 cos cp, sin (p).
Conversely, if v is a finite Bore! measure on [ -7r/2, 7r/2] and if for Borel A C 683 \ {0), M(A) is defined by the previous triple integral with x(9, cp) as before and B is diag(a, a, b), with a and b greater than 1/2, then M is a Levy measure, tBM = t - M for all t > 0, and A(M) c 0'.
[In the converse part of this theorem, we cannot conclude that B E f(µ) implies A(µ) = 0'. That is, µ may have an exponent which is diagonalizable even though A(µ) is discrete.]
Proof. By the assumed form of .9-(A(µ)), we have A(µ) = 0'. The first task is to construct v. For A a Borel subset of S. (cf. Theorem 4.3.7), let m(A) M{tBx: x E A, t > 1). Let = (D: SB x68+_ R3 {0) be given by
F(x, t) := t8x (cf. Proposition 4.3.4). Let ft be the measure on SB X68+ given by M := m X y, where dy t-2 dt on W. Clearly, M ='M. Next, we show that there is a finite Borel measure v on [ - r/2, 7r/2] such that m = A X v, where A is a Lebesgue measure on [0, 2w). Let R(0) denote the counterclockwise rotation about the z-axis through an angle of 0 radians. Let D and E be Borel subsets of [0, 27r) and [-vr/2, 7r/21, respectively. Since A(M) = 0', R(0)m(D x E) = m(D x E). Set aE(D) _= m(D x E), for E fixed. Then aE is a finite Borel measure on (0, 27r) which is invariant.under rotations, so OE =
OPERATOR-STABLE MEASURES
274
a(E) - A, where a(E) is a constant depending on E. But, a(-) as a function of E is a finite Borel measure on [-ir/2, it/2]. Set v(E) = a(E). Clearly, m = A X P. This establishes the stated representation for M. Now, we prove the converse of the theorem. We first show that M is a Levy measure. Clearly, M is a measure on R3 \ (0), so it remains to show that f(1x12 A 1)M(dx) < -, where c A d means the minimum of c and d. Clearly, for 0 < t 5 1, IItB11 < 3r", where a = a A b. Thus, f (1x12 A 1)M(dx)
f2irf
< kl + k2f r/2 -w/20
l() tBx(e, cp)
I2
A 1)t-2 dt d8 v(dcp)
o
for some constants kl and k2. But,
f(ItBx(o,)I2 o
A
1)t-2 dt < 9 f lt2("_') dt
1/2. Hence M is a Levy measure. The rest of the theorem follows easily.
Q.E.D.
4.16 BIBLIOGRAPHIC COMMENTS
The beginning of the study of the operator-stable laws presented in Section 4.2 is due independently to Sharpe (1969) and Sakovic (1961, 1964). Our presentation follows Sharpe (1969) and Hudson and Mason (1981). The material in Section 4.3 partially follows Hudson and Mason (1981), Jurek (1982c, 1984, 1989a), Krakowiak (1979), and Sharpe (1969). Section 4.4 is based on Jurek (1982c, 1984). The ideas
in Section 4.5 are due to Urbanik (1975, 1977, 1978). The structural
results and results on the exponents of an operator-stable law in Section 4.6 follow Holmes, Hudson, and Mason (1982), Jurek (1979a), and Sharpe (1969). Hudson, Mason, and Veeh (1983) is the basis for Section 4.7, whereas Section 4.8 is from Holmes, Hudson, and Mason (1982) and Luczak (1984). Material on the centering function is due to
Sharpe (1969). Also concerning b(t), Sato (1987) finds condition to eliminate it, that is, pe = t Bµ, when 1 is an eigenvalue of the exponent B. For Section 4.10 on the absolute continuity, see Hudson (1980).
BIBLIOGRAPHIC COMMENTS
275
The domain of normal attraction material is from several sources, Hudson, Mason and Veeh (1983), Jurek (1980), and Meerschaert (1990). Material concerning moments is from Hudson, Jurek, and Veeh (1986) and Meerschaert (1990). Section 4.13 on independent marginals follows Hudson (1980), Hudson, Mason, and Tucker (1981), and Veeh (1982). Also of interest is that there are nonstable laws on
R2 with all univariate marginals being stable [cf. Marcus (1983)]. However, if all two-dimensional marginals are infinitely divisible, then µ is stable if and only if all of its one-dimensional marginals are stable
[cf. Gine and Hahn (1983)]. The material on multivariate stable measures is well known [e.g., Levy (1937)], but also see Gine and Hahn (1983) and Dudley and Kanter (1974). The example following Levy's example is due to Linnik and Ostrovskii (private communica-
tion). Finally, when d = 3 is from Holmes, Hudson, and Mason (1982).
Epilogue
The selection of topics in this monograph was biased by our point of view. Nevertheless, we want to indicate some areas of research which
are close to the theory of operator-limit distributions and are not mentioned here. Operator-Semistability
Let S 1, S 21 ... be independent identically distributed random vectors, either R'-valued or Banach space valued, let A,, A 2, ... be bounded
linear operators, and let a1, a2, ... be vectors, as in Section 4.1. We say that µ is operator-semistable if there exists a sequence of natural numbers 1 1
as n - oo. Obviously, this yields the class of operator-stable laws whenever k = n, or for more general sequences with r = 1. The following is the main characterization of such p.. A full probability µ is operator-semistable if and only if there exist
c e (0, 1), an operator B, and a vector b such that
µ`=Bµ*S(b) [cf. Jajte (1977), Krakowiak (1979, 1980), Kruglov (1972), Luczak (1981, 1984), and Siebert (1986)]. In fact, the above equation was 277
EPILOGUE
278
studied in Levy (1937) for the real line. Similar to the spectral characterization of exponents of operator-stable probabilities (cf. The-
orem 4.6.5), there is a characterization of the operator B which appears in the above factorization of operator-semistable probabilities [cf. Jajte (1977)].
.Stability One way of looking at the Sakovic-Sharpe characterization is the following. Since t'A = µt * S( - bt) for all t > 0, we have
t1µ * sµ = (s + t)BA * 3(bt+5 - bt - bs) for all t, s > 0, where {t1: t > 0) is a group of linear operators. Thus, for an arbitrary group .# of bounded linear operators, we say that µ is
#stable if for each G G2 E A there exist G3 E .1 and a vector v such that G1µ * G2µ = G3µ * 8(v).
Taking f= {T.: a > 0), where Tav = av, this becomes the classical notion of stability of measure (cf. Section 4.14). Parthasarathy and Schmidt (1975) and Schmidt (1975) contain most of what is known in
the case Jc Aut(l '). Obviously, µ essentially depends on how large ,0 is. More precisely, µ is called completely stable when µ is Aut(lBd)-
stable. Parthasarathy (1973) showed that µ is completely stable, d >: 2, if and only if µ is Gaussian; also, see Mincer (1982a). In the case of an infinite-dimensional Hilbert space H and the group Aut(H), the situation is very different. The class of completely stable measures includes Gaussian measures and some non-Gaussian ones [cf. Mincer and Urbanik (1979)].
One-Parameter Group Normalization In Chapters 3 and 4, we dealt with the limit distributions of sequences
A»(f1 + ... +4») + a,
279
EPILOGUE
with sequences {An} from the full group Aut. However, the final characterizations of limiting µ were
12 =t8µ*3(b,),
t>0,
for operator-self decomposable and operator-stable measures, respectively (cf. Theorems 3.5.5 and 4.2.12). Hence, one sees that we could take A from a one-parameter group f tc: t > 0) = (e-s': s E If8) with
a fixed operator C. In infinite-dimensional linear spaces not all one-parameter groups are of exponential form (cf. Section 1.4). This leads to the following concept. For a given one-parameter strongly continuous group U of bounded linear operators on a Banach space X, we say that µ E L(U) if there exist U. E U and X-valued independent random variables ,, such that (i) U,.f1, 1 < j < n, n >_ 1, are uniformly infinitesimal;
(ii) Q f I + ... + n) + x
µ
for some x, (=- X. Similarly, p. E -,-(U) if the sequence (fn) is also identically distributed (cf. Sections 3.4 to 3.6, 4.3, and 4.4). The concept of normalization by elements from a group U was investigated in Jurek (1983a) and Kehrer (1983).
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INDEX
Absolute continuity, 162, 163 of exp(-Q)decomposable, 164, 173, 175
of infinitely divisible, 164, 165 of operator-stable, 237 Affine transformation, 24, 48, 59, 192 Araujd and Gine, xiii, 44,281
Convergence of types, 45, 48, 64, 65, 67, 68
Convolution, 23 Coordinate mapping, 41 Cramer theorem, 38, 58, 89 Cramer-Wold device, 26,269 Cuppens. 281
Aut(X), 4
Banach: dual, 3 space, 3, 67
Banach-Steinhaus theorem, 24 Billingsley, 44, 70, 281
Boundary, 20 Center of a group, 221 Centering function, 192, 236 Characteristic functional, 25, 32, 33, 57, 62,89, 101, 103, 119, 197, 264 Chorny, 281
Compact: conditional, 22, 25, 49, 50 sequential, 22 shift conditionally, 23, 25, 29 Continuity theorem, 26, 62 Continuous mapping theorem, see Weak
convergence, mapping theorem Convergence: of operator types, 52, 187, 202 of types, 45, 48, 64, 65, 67, 68
weakly, 20 Convergence of operator types, 52, 187, 202
de Acosta, 44, 281 deConinck, x, 281 de Miranda, 44, 285
Determinant, 6 Differentiable, 8 continuously, 8 Distribution, 22 converge in distribution, 22 Domain of attraction, 239 Domain of normal attraction, 239, 243, 247, 252
DONA, 240 Dudley, 275, 281 Dynkin, 44,281 Elliptically symmetric. 180, 229, 234 End (X), 4 Exponent: commuting, 219, 221 exp(-tQ)-decomposable, 89, 96 operator-selfdecomposable, 76 operator-stable, 185, 186, 192, 205, 211, 212, 215, 272
Exponential: exp(-tQ)-, 89, 101, 161, 162, 173 U-, 85, 179
289
INDEX
290 Fenchal, 70, 281 Fisz, x, 70, 281 Freedman, 183, 281 Functional, 24, 25
Gaussian, 31, 58, 88,90, 127, 142, 162, 213, 214, 246, 270 centered, 32, 33, 38, 39 moment, 252 GDOA, 240
Generalized domain of attraction, 250 Gikhman and Skorohod, 44,281 Gin6, 275, 281
Gnedenko and Kolmogorov, x, 281 Griffin, xii, 282 Haar measure, 181, 220, 223, 231 Hahn, xii, xiii, 275, 281, 282
Hausdorff measure, 230 Hazod, xiii, 282 Hilbert space, 3, 26, 62 Hille and Phillips, 24, 44, 82, 282 Hoffman and Kunze, 44, 282 Holmes, 274, 275, 282 Hudson, viii, xi, 274, 275, 282
Ideal, I Idempotent, 1, 2, 55, 77, 78 ID(X), 28, 34, 35 IDIae(X), 36, 124, 127, 128, 132, 149, 168, 173, 198, 199, 201
Ilinskii, 67, 282 Infinitely divisible, 28, 29, 240 absolute continuity, 163, 165 Infinitesimal system, 35, 38, 72, 84
Integrable Bochner, 8 Integral: random, 117,120 representation, 116 Invariant: semigroup, 50,65 space, 6, 63
Kehrer, 279, 284 Klass, xii, xiii, 282 Klosowska, 284 Krakowiak, xiii, 274, 277, 284 Kruglov, 277, 284 Kucharczak, xiii, 284 Kumar, 183, 284
Laha,284 Lamperti, x, 284 Lange, 180, 284 Lesvy, 269, 275, 278, 284 class, 71
-Kintchin representation, 33, 101 spectral measure, 31, 33, 34,95 Lie: bracket, 15, 17 tangent space, 15, 85, 189, 212, 271
theory, 15 lin(A), 105 linQ(A), 105 Linde, x, 284
Lindvall, 284 Luczak, 274, 277, 284, 285 Lukacs, 237, 285
Mandrekar, 284 Marcus, 275, 285 Marczewski and Ryll-Nardzewski, 285 Marginal, 254 complete, 254 independent, 254, 255, 259
k-dimensional, 254 orthogonal, 254 Markov process, 144, 150 Mason, x, 274, 275, 282, 285
Measurable function, 7 Measure: decomposable, 72 equivalent, 162 regular, 20
Inverse function theorem, 9
Meerschaert, 275, 285 Michalicek, 285 Mincer, xiii, 278, 285
J, metric, 41
Minimal polynomial, 6,205
Jajte, xiii, 277, 278, 283
Moment, 249, 253, 270 exponential, 32, 35
Jurek, 44, 70, 182, 183, 274, 275, 279, 282. 283
Kanter, 275, 281
Gaussian, 252 logarithmic, 96, 103, 122, 124, 127, 130, 161,179,198
291
INDEX
Monothetic semigroup, 2, 3, 55
206, 210, 215, 253
Prohorov: Niedbalsha-Rajba, xiii, 67, 285 Nobel, 282 Norm: new, 92
operator, 4 Norming sequence, 72, 74, 201 Numakura, 44,285 theorem, 1, 2, 55, 76, 77, 80, 81
One-parameter semigroup, 10, 71, 76,
metric, 22 theorem, 22, 23
Random integral representation, 71, 178
Resolvent equation, 145 Riesz representation theorem, 4 Rohatgi, 284 Rolle's theorem, I l l Rosinski, 183, 284
122, 124
contraction, 10, 13, 147, 152 operator, 144 strongly continuous, 10, 12, 13, 14, 78. 83,279
Operator: bounded, 4 closed, 5, 11
contraction, 10 convergence of types. 52 core, 14 covariance, 33 extension, 5 generator, 10, 11, 12, 13, 14,108,116, 142, 144, 149, 152
graph, 4 log of, 17, 82 S-, 33, 37, 88, 90 -selfdecomposable, 71, 72, 74, 76, 78, 83
-semistability, 277 -stable, 185,191 topology, 24 trace, 6 type, 51
Ornstein-Uhlenbeck, x, 149, 150, 161, 183
Orthogonal transformation, 18, 59, 60, 61
Paalman, 44, 285 Parthasarathy, 44, 278, 285
Sakovic, x, 186, 274, 285, 286
Sakovic-Sharpe characterization, 186 Sato, x, xiii, 183, 274, 286 Schmidt, 278, 285, 286 Schreiber, 183, 284 Schur's lemma, 180, 181, 230, 272 Selfdecomposable, 177 sem(a), 2, 55, 75, 76, 77 Semigroup: decomposable, 45, 53
Feller-Dynkin, 152 full, 46, 47, 63
invariant, 50, 65 Semovskii, 286 Separable, 23, 39 Sharpe, xi, 70, 186, 274, 286 Siebert, 277, 286 Skew-symmetric transformation, 18, 88, 213, 229, 271
Skitovich theorem, 38, 58, 286 Skorohod: metric, 41, 42, 43 space, 39, 42, 43, 117 Smalara, 44, 284 Spectral: function, 94, 109, 112 measure, 31 Stable: multivariate, 261, 263, 265, 266, 268 strictly, 261 1B, 193
Pollard, 44, 285
Stationary, 161 Stochastically equivalent, 40 Subadditive, 35, 36 Submultiplicative, 35, 36
Portmanteau theorem, 20 Primary decomposition theorem, 6,205,
Support, 26, 27, 28, 38, 105, 106, 197, 205, 270
Paulauskas,285 Poisson measure, 30 Polar coordinate, 92, 110
INDEX
292
Symmetric: elliptically, 234 group, 4S, 53, 59 semigroup, 53 Symmetrization, 26, 29 Tangent space, 15, 85, 189, 212, 271 Tight, 20, 22,49 Topological:
semigroup, I subsemigroup, I Trace, 6 Tucker, xiii, 275, 282, 286
U-exponent, 85 commuting, 88 Uniformly asymptotically negligible, 35 Urbanik, xiii, 44, 45, 67, 70, 178, 182,
274, 278, 285, 286, 287
decomposability semigroup, 45, 53, 59, 66, 67, 71, 76, 83, 108
Veeh, viii, 274, 275, 282, 287 Vervaat, 183, 284
Watanabe, 44, 287 lemma, 14, 160 Weak convergence, 20, 38 mapping theorem, 21, 24, 86 Weissman, 70, 287 Williams, 183, 287 Wolfe, 182, 183, 287 Yamazato, xiii, 183, 286, 287
Zolotarev, x, 287
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