ON A DIFFERENCE EQUATION ARISING IN A LEARNING-THEORY MODEL BY BERNARD EPSTEIN* ABSTRACT
An analysisis presented of the...
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ON A DIFFERENCE EQUATION ARISING IN A LEARNING-THEORY MODEL BY BERNARD EPSTEIN* ABSTRACT
An analysisis presented of the equation f ( x + a) -- f ( x ) = e - x {f(x) -f.(x--b)}. Here a and b denote arbitrary positive constants, and a solution is sought which satisfiesthe followingconditions: f(-- ~ ) =0, f( + cx3)= 1, 0 =3q O.
It is immediately evident that if fo(x) satisfies the boundary conditions, the same will be true of all the succeeding functions fl(x), f2(x),-". The monotonicity and the positivity of the factors w(x) and 1 - w(x) are readily seen to imply that, if fo(x) is monotone, all the succeeding functions will also possess this property. Finally, the positive character of the factors w(x) and 1 - w(x) evidently guarantees that if fo(x) is so chosen that fl(x) > fo(x) holds everywhere, then, more generally, fn+l > f,(x) will also hold everywhere; similarly if the inequalities are reversed. Now suppose that we can find monotone functions, fo(x) and go(x), both satisfying the boundary conditions, such that the inequalities f o ( x ) < f l ( x ) , go(x) > gl(x) hold everywhere. Then it is immediately evident that the sequences fo(x),fl(x),"" and go(x),gl(x),.., converge pointwise to monotone functions, f(x) and g(x) respectively, which satisfy (1). If, furthermore, fo(x) < go(x) holds everywhere, it then follows immediately from (5) (and the analogous equation for the g's) that f,(x) < gn(x) everywhere. Thus, f(x) < go(X), and so (6a)
0 f ( +
c~) >__f0( + ~ ) = 1.
Thus, f ( x ) satisfies (1) and the prescribed boundary conditions; similarly for g(x), and so, by the introductory remarks, f ( x ) = g(x). It remains to demonstrate a pair of functions {fo(x),go(x)} satisfying the conditions imposed in the preceding paragraph. Simple calculations show that the following choices will suffice, provided a > b:
-
(7a)
fo(x) =
{
2
k(k -
2k(k
1)
/
1)} '
na<xl;
2a log 2 b
x > --7.
go(x) = l 1,
(The aforementioned restriction, a > b, is immediately set aside by the following observation: If the unique solution which has been shown to exist for a = b is denoted as f ( x ; a , b ) , then 1 - f ( x;b,a) satisfies (1) and the boundary conditions when b > a. Therefore, we may confine attention to the case a > b.) Thus, we have demonstrated the existence of a solution to the given problem, and it follows from (7a) and (7b) that the boundary values are approached with Gaussian rapidity: (8a)
f ( x ) = O(exp { - x2/4a}),
x ~ - oo,
(8b)
f ( x ) = 1 - O ( e x p { - 1 -e)x2/2a}),
x~ + ~.
3. We now proceed to obtain an analytical expression for the solution whose existence has been demonstrated in the preceding section. The rapid approach of f(x) to its limiting values at + ~ , as indicated by (8a) and (8b), guarantees the existence, for all values of the complex parameter s ( = o" + it), of the bilateral Laplace transform
(9)
f?ooe -~:'{f(x) - u(x)} dx.
148
[September
B. EPSTEIN
[Here u(x) denotes the "unit step-function" 1/2(1 + sgnx).] Therefore, the integral (10)
f(s) =
x)e-'Xdx
exists for a > 0 and may be analytically continued to the whole (finite) s-plane except for a simple pole of residue + I at the origin. From (1) one readily obtains, for a > O,
](s){e °~- 1} = f(s + 1){1 - e-a(~+l)}, or 1 -- e -b(s+l)
(11)
f(s) = f(s + 1)e -"~
1-
e -as
To eliminate the factor e -"~ , let
(12)
f(s)=exP { 2 (s2-s)} } g(x).
Then from (11) and (12) one obtains 1 -- e -b(s+l)
g(s) = g(s + 1)
1 -- e -as
'
and more generally, for any positive integer k,
(13)
1 -- e -b(s+k) g ( s + k --
1) = g ( s + k ) 1 - e - a ( s + k - t ) "
Writing out (13) for k = 1,2,...,n and then multiplying and simplifying, one obtains
g(s) = g(s + n)
1 -- e-a(s+n)
1 - e -o~
fi {1 - e -b("+k)}
k=l
fi
{I - e -"c~+~ }
k=l
(14) Letting n ~ oo, one easily sees that the two products appearing in (14) converge for all s, not only for tr > O; it follows that g(s + n) converges to an entire function, which will be denoted as h(s). Since h(s) = l i m ~ ~og(s + n) = lira..., oog(s + (n + 1)) = lim~_,® g((s + 1) + n), it follows that
1966]
DIFFERENCE EQUATION IN LEARNING-THEORY MODEL
(15)
h(s + 1) = h(s).
149
Thus, from (14) one obtains
(16)
f l {1 - e -~(~+k) }
h(s) g(s) = i -
k=l
e -~
f i {1 - e -°~+k) } k=l
and hence co
e ("/2)~2 h(s) k~l {1 - e-bO+k)} (17)
f(s) =
2 sinh as~2 f i {1 - e -"(s+~')} k=l
Now, the infinite product appearing in the numerator in (17), which we shall henceforth denote by b(s), possesses simple zeroes at the points s = - k + (2rmri/b) (k > 0, m ~ 0), while the other product, which will be denoted a(s), has simple zeros at s = - k + (2mzi/a) (k > 0, m ~ 0). Thus, the quotient of these products is analytic and different from zero at the negative integers, but possesses simple poles at the points s = - k + (2mrci/a) (k > 0, m # 0). (Actually, this assertion is justified only if b/a is irrational, for otherwise some of the non-real zeroes of the numerator and denominator cancel each other out; nevertheless, the final result is easily seen to be valid for rational as well as irrational values of b/a.) Since f(s) must be analytic everywhere except at the origin, the factor h(s) must possess zeroes at the non-real zeroes of a(s) and of sinha2/2; i.e., at the points - k + 2mrri/a (k > 0, m # 0). From the periodicity property (15) it then follows that h(s) must vanish at the points - k + ( 2 m r c i / a ) ( k ~ O , m ~0). Now, the theta-function 01(ns, exp ( - 21rZ/a)), which for brevity will henceforth be denoted O(s), is entire and possesses simple zeroes at the p o i n t s - k + 2mrci/a (k ~ O, m ~ 0); furthermore, (18)
O(s +
1) =
-
O(s).
Hence, the entire function O(s)/sin zs has simple zeroes at precisely the points which have been shown to be zeros of h(s), and it is periodic with period one. It follows that
O(s) (19)
h(s) = finns H(s),
where H(s) is also an entire function with period one. From (17) we now obtain
September
150
B. EPSTEIN
(20)
H(s) = a(s) 2 sinh as/2 sin zcs ~ b(s) exp(as2]2) ~(s) f ( s ) .
It will be shown next that H(s) must reduce to a constant. For all values of s, the inequality (21a)
12sinha"2sin '] _ Cexp { 2n2n+a2n2n2 } '
10(s) l
where C denotes the positive minimum of on the line t = rc/a. Finally, since f(x) is everywhere positive, it follows from (10) that the inequality
lf(s)l
(25)
holds everywhere in the half-plane a > 0. Taking account of these several inequalities, one finds that, on the line segment N < a < N + 1, t = (n(2n + 1)/a), the inequality (26)
IH(s) I < =
C' exp 2rc2n a
holds, where C' = (4/C)exp(3rc2/2a). By periodicity, however, the above restriction N < tr < N + 1 may be dropped, so that (26) holds everywhere on the line t = n(2n + 1)/a. Now let s be confined momentarily to real values. As a smooth periodic function, H(s) can be expanded in a convergent Fourier series:
1966]
DIFFERENCE EQUATION IN LEARNING-THEORY MODEL
H(s) =
~, Cke 2*i*', Ck = ~l H(s)e
151
ds.
do
k=-oo
Employing the Cauehy integral theorem, one obtains (28)
Ck =
fr
H(s)e - 2,ak~ds,
where Fn denotes the path consisting of the vertical and upper sides of the rectangle with vertices at 0, 1, ~(2n + 1)i/a, and 1 + (~(2n + 1)i/a). By periodicity, the integrals along the vertical sides cancel, and one is left with the integral along the upper side of the rectangle. Taking account of (26), one obtains (29)
--a--/ + n(2k + 1)]
I ck I < C' exp
Since n may be chosen arbitrarily large, it follows from (2) that ck must vanish for negative values of k. For positive values of k the same result is obtained by integrating in the lower half-plane. Therefore, H(s) must reduce to a constant on the real axis, and hence, as asserted above, everywhere. The value of this constant is determined by the condition, stated after (10), that f(s) must have a residue of + 1 at the origin. In this way we are led to the formula
(30)
C" expas2/2 0(s) b(s) f(s) = 2 sinh as~2 sin rcs a(s) '
where (31)
rra Cp
__
_
a(0)
_
0'(0) b(0)"
Thus, the function f(x) whose existence was demonstrated in the previous section must admit the integral representation (32)
1 fo°+i~e~Xf(s)ds, y(x) = 2rri ~oo-,oo
ao>0.
Alternatively, one could have obtained (32), independently of the considerations of the preceding section, simply by assuming the existence of a solution to the given problem whose behavior near _+ oo permits the use of the analytic device which we have employed. It is then quite simple to justify this procedure a posteriori by showing directly that (32) defines a continuous functionf(x) satisfying (1) and the prescribed boundary conditions; as might be expected, the proof that the boundary conditions are satisfied involves a suitable change in the path of integration and an application of the Riemann-Lebesgue lemma. (In fact, f(x) is easily shown to be analytic in the complex variable x in a strip of width 2rr about the real axis, and to depend continuously on the parameters a and b.) However,
152
B. EPSTEIN
we have been unable to establish directly from (32) either the monotonicity off(x) (or even the inequalities 0 - ~p(z"). In conclusion, we establish a lemma which we used in the proof of Theorem II.
LEMMA 6. When l is even, a boundary point f of P(Z) belongs to that cone if and only if H is satisfied. Proof. We suppose l = 2N. As before, we deduce a contradiction from the hypothesis that f is a boundary point of P(Z) which does not belong to that cone but which does satisfy II. Our argument is much the same as before; f is the limit of a sequence f~ in the interior of P(Z), and these functions may be represented by rational functions of degree N in P[zl, zl]. Since there is a choice for these representatives, we take them in such a fashion that the nearest poles are distributed symmetrically about [zl, zt]; more precisely, we select each time the representativefttn)for f~ so that if 2' is the nearest pole offt°)to the left of zl and if2" is the nearest pole to the right ofz~, then z~ - 2' = 2" - z,. A subsequence of the sequence of rational functions so determined then converges to a rational ~b([) which coincides with f at all points of Z except, perhaps, the end points. The representatives have been chosen in such a way that at least two poles are destroyed, i.e. ¢(() is necessarily of degree at most N - 2. Thus the Loewner determinant for ¢(~) associated with the set z2, z3, z3,'", z~_ ~ vanishes and therefore the corresponding Loewner determinant o f f does. We now argue exactly as in the proof of Lemma 3 to infer thatf(zt) = ~b(zl) andf(zl) = ~b(zs), hence t h a t f is in P(Z), a contradiction. 4. The cone P'(Z). By P' we denote the subclass of P consisting of functions which are regular and positive on the open right half-axis. These functions admit the canonical representation obtained from (1) which follows:
(3)
0o
where • _~ 0, /~ = ~b(0) > 0 and j'_°~(1 + ~2)-1d/~(2) is finite. It is easy to see that if q~(~) belongs to P', so also does 6(~)= [~b(1/~)]-1 as well as ¢*(~) = ~¢(1/£). In this section we shall suppose that Z is a subset of the open right half-axis and shall seek necessary and sufficient conditions that a function f i n C(Z) should
1966]
THE THEOREMS OF LOEWNER AND HCK
167
belong not just to P(Z) but to P'(Z), the cone consisting of restrictions to Z of functions in P'. It is clear that the cone P'(Z) is closed, for a sequence in P' which converges on the points of Z has a subsequence converging on all points of the positive real axis, those points being bounded away from the supports of the measures; moreover, the limiting function is non-negative on the right half-axis, hence is in P'. It is also evident that i f f belongs to P'(Z) there exists a non-negative value C such that i f f is extended to the origin by f(0) = C, the extended function is in P(Z u 0). Unfortunately, we cannot always take C = 0. We introduce the set Z* consisting of reciprocals of points in Z
1/zl < 1/z1-1
< 1/zt-2
< ""
< 1/Zl
which may also be written z* < z* < z* < ... < z* and consider the following conditions, concerning f in C(Z). III. f may be extended to a non-negative function in P(Z u O) IV. The functionf* defined on Z* byf*(z~) -----z~f(1/z*) may be extended to a non-negative function in P(Z* u O) V. The function J~defined on Z* by j~z*)= [f(1/z*)] -I may be extended to a non-negative function in P(Z* u 0). We then have TrmOREM III. A function f in C(Z) belongs to P'(Z) if and only if (a) when l is odd, 111 and V are valid (b) when l is even, 11I and IV are valid. Proof. The necessity is an immediate consequence of our comment concerning the functions ~ ( 0 and ~b*(O when ~b(() is in P'. For the sufficiency we must give different arguments depending on the parity of L We remark that we possess examples showing that the state of affairs is essentially different when I is odd and when l is even. When l = 2N + 1 is odd, we pass from f in C(Z) satisfying III and V to its canonical representation ~b(0, a rational function of degree at most N belonging to P[zl, zd. Since there exists a non-negative function in P[0, zl] which coincides with f on Z, it follows from Lemma 5 that ~b(0 is regular and non-negative in [0, zd, and we have only to show that this function has no poles to the right of z~, thereby putting it in P' and therefore putting f i n P'(Z). However, we may argue similarly with the functionfdefined on Z* to find that its canonical representation ~ ( 0 is non-negative and regular in [0, z*]. Since both ~b(0 and ~k(0 are rational and of degree at most N and satisfy the equations ~b(z*)= [~b(1/z*)]-1 for all 2N + 1 values of k, it follows that identically in ~ we have
~#(0 -- [~(l/O] -1.
168
W.F. DONOGHUE, JR.
[September
The regularity of ~(~) in [0, z*] therefore implies the regularity of qS(~) in z z < z < + oo. Hence ~(~) is in P'. When l = 2N is even our argument is somewhat more complicated. Since III and IV surely imply that f is in P(Z) and f * is in P(Z*) we will suppose at first that each of these functions is an interior point of the corresponding cone and make use of the representation theory developed in the previous section. Let C be so chosen that w h e n f i s extended to 0 by the definition f(0) = C the extended function is in P(Z u 0); because of III there exists such a C which is non-negative. In the familyft(z) associated w i t h f w e select the functionf,(z) for which f~(0) = C; this function is rational and of degree at most N and is the canonical representation of the extended function considered on the l + 1 points of Z u 0. It follows that f~(z) is in P[0, zt]. It is not difficult to see that if t is varied so that the poles o f f t move to the right, the numberft(0) diminishes; it follows that we can pass continuously to that member of the family for whichft(0) = 0 without departing from the class P[0, zt]. We let t = 0 correspond to the rational function so determined; fo(z) is of degree at most N and belongs to P[0, zz] and satisfies fo(0) = 0. Since fo(z) is non-negative in [0, zl] we have only to show that it has no poles to the right of z~ to make sure that it belongs to P'. For this purpose we pass to the ratonal function g(~) = ~fo(1/~) which is also of degree at most N and which coincides with f * on the points of Z*. It follows that g(£) is a member of the familyft*and therefore that the residue ofg(~) at any pole to the left of z* = 1/z~ is negative. If, now, fo(~) had a pole to the right ofzz, g(~) would have one in the interval 0 < z < z* and the residue there would be negative. However, if we make the explicit computation we will have fo(~) = h(~) + ~
m
where m > 0 and ~ < z t with h(~)
regular near 2, and g(¢) = CY(ll¢) +
-
which has a positive residue at the pole 1/2. Thus fo(~) had no pole to the right of z, and was therefore in P'. Finally, if the functions f and f * are not both interior points of their respective cones, we have only to pass to f + e x/z which corresponds to f * + e x/z for small positive ~. The perturbed functions are interior points of those cones and also satisfy III and IV. From the fact that P'(Z) is closed we infer that f i s inP'(Z), completing the proof of Theorem III. We do not give the easy proof that an element f i n P'(Z) may be extended to the origin by f(0) = 0 if f is an interior point of P(Z).
1966]
THE THEOREMS OF LOEWNER AND PICK
169
5. The theorems of Pick and Carath~dory. The problem considered in Section 2 can equally well be studied under the hypothesis that the finite set Z is a subset of the open upper half-plane; we would then seek conditions for a function in C(Z) to belong to the cone P(Z), the restrictions to Z of functions in P. The solution has been given by Pick. [5]
THEOREM. A function f(z) in C(Z) which is not a real constant is the restriction to Z of a function c~(~) in P if and only if the imaginary part off(z) is positive and the matrix of order l
Pij
--
f(z~) -- f(zj) z i -- ~j
is a positive matrix. This matrix has the eigenvalue 0 with multiplicity k > 0 if and only if qb(~) is a rational function of degree I - k and in this case ~(¢) is determined uniquely by the data. We do not give a proof of this theorem which can be established by the same arguments which we have used to prove Theorems I and II, the proof, however, is substantially easier since in the present case the cone P(Z) is closed and we may also always argue with positive matrices rather than with determinants. A completely analogous theorem is due to Carath6odory who considered the convex cone of functions u(z) harmonic and positive in the unit circle; such functions admit a Fourier expansion
u(r e i°) = Z
ckrlkl eikO
the summation being taken over all integers. The following theorem is due to Carath6odory. TI-mOREM. A system of numbers {el} - N = 6(s')
Also, as is readily verified, Received June 13, 1966. * I am indebted to Dr. Micha Perles for these references. 171
i72
M. EDELSTEIN
(3)
[September
6(e)< T
LEMMA X. Let x, y e V, x # y, and suppose 0 < # < ½ ; (4)
then
X - I ] #x + (X - #)y It > 2#6(11 x - y [I)
Proof.
Let z = #x + (1 - #)y. It clearly suffices to show that all w ~ X with II w - z II --< 2#6(11 x - y I1~ are in V. Set v = ½ #(w - (1 - 2#)y). Then w = 2#v + (1 - 2#)y is a convex combination o f v and y and it suffices to show that v is within distance 6(I [ x - y I[) from ½(x + y). Now X IIv - -~ (x + y)ll
=
1 [ l w - ( 1 - 2 # ) y - #(x + 2---#-
y)[t
X 2# IIw - #x - (x - #)y II 1 2# II w - z LEM_V~ 2. L e t 0 < ~ < X , the following conditions
0 1 . (.x.+ l _ - c.
> 1 - 6"(1) and, it clearly follows from (1), that [[ x , + t -- Xn[[ < R,5 -1 (1).
Xn - - Cn
1966]
FARTHEST POINTS OF SETS
175
Thus (setting e~= 6 n+l (1)) we obtain
f.(x.+ 1 - c.) > Rn(1 - 6~(1)) as asserted. Let now s = lim..oox, and suppose c = lim._.oo c.. We clearly have
sup {lie - x II Ix ~ s} -- lira (sup {11e,- x II Ix ~ S}) n--~ O0
lira r~+l = lim [Jc.-x~+ll]=l[ c - s l l M-~O0
N --~ O0
concluding the proof of the theorem. REMARKS. In [5] Lindenstrauss defined the notion of a strongly exposed point as follows: A point s e S is said to be a strongly exposed point of S if there is a n f e X * such that f ( y ) < f ( s ) for y 4 s and whenever {xn} c S is such that f(xn) ~ f ( s ) t h e n Since every point on the boundary of the unit ball of a uniformly convex Banach space is known to be strongly exposed it follows from the above theorem that every closed and bounded set in a uniformly convex Banach space has strongly exposed points.
IIx,- sll-.0
3. DEFINITION. A normed linear space X is said to have property (I) if every closed and bounded convex set in X can be represented as the intersection of a family of closed balls. This property was introduced by Mazur I'6] and shown to hold for all reflexive Banach spaces having a strongly differentiable norm (cf. also Phelps 1'7, p. 976]). THEOREM 2. Let X and S be as in Theorem 1 and suppose, in addition, that X has property (I). Then S = 66 b(S). Proof. Clearly ~-6b(S)c~-6S. To prove the reverse inclusion suppose x (~c-6b(s). Then, by property (I) there is a closed ball
B(co, r) = {yl IIy-coil ~ r}, where c o e x and r > 0 , such that ~-6b(S) cB(co, r) and x - c o I l - r > 0 . By Theorem 1 there i s a c e X such that [ c - c o < X - C o - r with cEC. If s e S is farthest from c then I I s - c l l < s - c o l + Co " c < [[x - c o showing that S c B(co, r). Thus x ¢ c-6S and ~-6S c ~"6b(S) completing the proof.
176
M. EDELSTEIN REFERENCES
1. E. Asplund, ,4 direct proof of Straszewicz' theorem in Hilbert space (to appear). 2. - - , The potential of projections in Hilbert space (to appear). 3. M. Edelstein, On some special types of exposed points of closed and bounded sets in Banach spaces, Indag. Math. 28 (1966), 360---363. 4. V. L. Klee, Extremal structure of convex sets II, Math. Z. 69 (1958), 90-104. 5. J. Lindenstrauss, On operators which attain their norm., Israel J. Math. 1 (1963), 139-148. 6. S. Mazur, Ober schwacheKonvergenz inden Raumen (LP), Studia Math. 4 (1933), 128-133. 7. R.R. Phelps, A representation theorem for bounded convex sets, Proc. Am. Math. Soc. 11 (1960), 976-983. SUMMER RESEARCHINSTITUTE, CANADIAN MATHEMATICALCONGRESS DALHOUSIE UNIVERSITY, HALIFAX, NOVA SCOTIA
PROPERTIES OF GENERALISED JUXTAPOLYNOMIALS BY
YEHORAM GORDON*
ABSTRACT
Given F(z),fl(z),f2(z) . . . . fn(z) defined on a finite point set E, and given B - - the set of generalised polynomials ~,= 1 a~fk(z) -- the definition of a juxtapolynomial is extended in the following manner: for a fixed 2(0 < ). __ 0, 8 + 4' < 1, such that f(z) E J1(2' + 8,E), because otherwise for every e > 0, e + 4' < 1, exists g,(z) e B with [ g,(z) - F(z) l < (4' + 8)If(z) - F(z)[ on E, and similarly as above taking a sequence 8m~ 0 with g,,,(z) converging to a limit function g(z) e B on E, we obtain the inequality ] g ( z ) - F(z)[ < 4' If(z)- F(:)[ on 6, impossible by our assumption. Hence f(z)~ Jz(2',E). Note that the classes which compose B are all distinct. We see now that f ( z ) e Jz(2,E)¢~. 2 < 4'. We shall show later how 4' may be calculated under certain conditions (Theorem 4). Lemmas 2, 3 are required in the proofs of some later results. 5.
LEMMA2.
the inequality
Let f ( z ) e B and 0 < 2
IF(g°) -f(z°)] (2 - I F(zo) -
g('o)
IF(zo)-:(zo)l
IF(zo)- g(zo)[ = IF(go) -f(zo) + h(zo) l,
thus (1) holds for every z e E. / K[~ ~ Conversely,(1) (with,~ = 1)impliesthatwhenF(zo)~f(zo) , Re '(')f(J~-~---°)F(zo) ._v.-
\
>0.
Let a__sup{I
h(z).12/2
:(z)--F(z)
and put g(z) -f(z)
(h(z))
Re,:(z) - r(z) ;~E,:(z)~ r(z),,
Ih(z) + a' then inequality(2) easilyfollows.When, for Zo ~ E,
F(zo) =f(zo), then h(zo)= 0, hence g(zo)= F(zo), and again (2) is satisfied. i If 2 < I, let h(z)- 1.d1__~(f(z )- g(z)) be the relation between g(z)EB and h(z)~ B. Obviously,
]r(~o)-g(zo)] ~: ~lF(zo)-f(Zo)] ,~ (I) holds
for go.
Note that I.emma 2 holds if 0 < 2 < 1 and we replace above "JI(2,E)" with "J2(~,E)", and replace " < " with "_- 0 (0 < i < m) and wj (1 < j < l), such that (3)
m ! ]~ 2if~(zi)(F(zi)--f(z~)) + ~, wJk(c~) = 0 i=o
j=t
for every 1 < !¢ < n. (ii) I f (3) holds for some such Era+l, 2~, wj, then f(z)¢Jl(1,Em+ 1 UEo).
Proof. We retain the notations of Theorem 2. (i) By Theorem 2, f ( z ) ~ J l ( 1 , E m + t UEo) for some such Era+ 1. By Lemma 3, a ~ H ~ R ( E m + t ) ~(a, 1), that is CR(Em+I)t~ H(a) # ~ , meaning CR(Em+ 1 ) n H J # ~ (where CR(Em+I) denotes the convex hull of R(Em+I), and H ± the space orthogonal to H). Therefore there exist constants Z t ~ 0 X t % o ~ q = l , real constants g~,]~j (1 ~ j _- 1. 2' > 0, since 2' = O=~F(x)~Bt (i) =~: Let 2 ' = 2 ( E ' + 1 ) = 1 , using (4) and (5) we have p(x;)/to'(x't)~_O, and from d2 f(x)~Jx(1, E'+I) c_ J1(1, E). (i) ~=: By cl there exists E~+ 1 such that f(x) ~ Jx(1, E'+ 1), using (4) and d2, (5) follows with equality everywhere, giving 2(E'+1)= 1. (ii) =~ : Suppose 2' = 2(E'+ 1) andf(x) ~ J1(2', E'+ i), then there exists g(x)~ B such that I g(x;) - F(x;) l ~ 2' If(x;) - t(x;) I i = 0,1,.--, n. Let (4) relate r(x) =-x'+ ... and g(x) for E'+I, then
1=
r(,:,)
I
to'(xO -< i~= 0
i=0 N
f
,=o
to'(x9
=
!
F-~--f(-~(~)
< 2(E~+1) ,=o
F(xl)-f(O(x't)
this contradiction implies f(x) ~ Jl(2', E'+ 1) _c J1(2', E). For any E,+l={xo,...,x,}, f(x)~J2(A(E~+l),E~+l). This is obvious if A(E,+ 1) = 0. If 2(E,+ 1) > 0, let (4) relate p(x) - x n+... andf(x) on E,+ 1, define
r(x)-
~ 2(E.+,)IP~X') l=0
] to(x)
~
X ~
Xl ~
where t o ( x ) - f i ( x - x , ) /=0
then I r(x,) I = 2(E, + 1) I P(x~)l, meaning f(x) ~ J2(2(E~ + 1), En + 1) ( ~ J2(2', E,+ 1)). Let
c(x) = {(al,
8";
i=l
a,f,(x)- r(x) I rli(x)- r(x)l},
_n we have [ 7 C(x~)~ ~ for every En+l ~ E, and by Helly's theorem (the sets l=O
C(x) are convex) N ~eE C(x) ~ ~ , so f(x) ~ J2(2', E). Therefore f(x) ~ Jz(2', E). (ii) ~ : JI(1,E) = J2(1,E), therefore 2' < 1. COROLLARY 1. Suppose f(x) = F(x) only on Ek = {Xo, Xl, "",x~-l} where 0 3, let xu ~ C(F) n n3 and assume that A' = {Xo, x~,..., Xn- 1} contains x , x j, Xk, X~, let the plane n contain A', and let F' = {x~j;0 < i ~ j < n}, n ' = lr, n n r = 1,2,3. Reducing the problem to the (n - 1) dimensional "space" n, the proof is carried out by induction. STATEMENT C: With the notations of statement b, dora, n2) d(na, nl)
d(xtj , x~) d(x+j, x j)
1+ 2 1 2"
0 such that, if Q:l~m-->B and R : I ~ - - , B are linear maps, and if
1, then there is a chain of subspaces E~ = Fo ~ F t c . . . F,~. 1-,~ = E~÷ 1 such that F, is isometrically isomorphic to
196
E. MICHAEL AND A. PELCYZNSKI
[September
1~+, (v = 0, 1,2, ..., nk+X- nk). But the existence of such a chain of subspaces follows immediately from Lemma 3.2. (iii) ~ (i). This implication is trivial. 5. A refinement of Theorem 1.1 for B = C(S). In the case of a C(S) space, S compact metric, one can prove a slightly stronger result than Theorem 1.1 (cf. Corollary 5.2 below). We recall that a finite-dimensional subspace E of C(S) is called a peaked partition subspace provided it is spanned by a peaked partition of unity, i.e. by non-negative functions fx,f2,'",f, such that Zfi = 1 and I]f~ll = 1 (i = 1,2, .-. n), where 1 denotes the function which is identically one on S. PROPOSITION 5.1. Let E be a linear subspace of C(S). Then the following conditions are equivalent. (o) E is a peaked partition subspace. (oo) E is isometrically isomorphic to l~ for some n, and 1 e E. Proof. (o)--*(oo). This implication is well known (cf. e.g. [9], [11]). (oo)-~ (o). Let f~'e E correspond under some isometric isomorphism to the k-th unit vector u~n) of l~ for k e n . Then clearly [[fkll = 1. Let us set k : [/[(s~)] t - l v j'~, where sk e S is chosen in such a way that [f~(sk)[ = 1. Clearly (cf. Lemma 2.3 (c))f~(sz)= 0 for l # k, l e n. Since l e E , there are scalars (t°)k , , such that 1 = ]~=lt~f~.Thus 1 = l(sz) = Y.~=lt°fk(s~)=t ° for l e n . Thus oo S ~ s n t ' Zk=lfk() 1 for s e S . Since [] Zk=lt~fk]l=]] Zk=lkf~]]=maxk~,]tk] for arbitrary scalars tl, t2, ..', t,, we get ~=l]fk(s)] < 1 for s e S. Thus all fk are non-negative. Hence {A,A,'",fn} is a peaked partition of unity. Combining Proposition 5.1 with the main result of [9] and Lemma 3.2 (cf. the proof of implication (ii)-, Off)) we get COROLLARY 5.2. Let E be an m-dimensional peaked partition subspace in C(S), S compact metric. Then there exists an increasing sequence of peaked partition subspaees E 1 ~ E2 ~ ... of C(S), whose union is dense in C(S), such that dimEn=n for all n and Em= E. 6. Monotone bases in separable ~]o -spaces. We recall (cf. [3, p. 67]) that a sequence (e,),~ t is called a (monotone) basis for a Banach space B provided each b in B has a unique expansion b = ] ~ , t~ei(and ]] b ]}~ [} Z=x t,e~ ][for n = 1,2,...). If (e,)~=x is a monotone basis for B, then the operator P~b = Y~--_ltiei, for b = Y-~=ltie~ e B, is a projection of norm one from B onto the subspace En spanned by et,e2,-.',er Therefore the existence of a monotone basis in B implies the existence of projections P,: B ~ E~ such that (a) liP-I[-- 1, (fl) each range P,B = E, is an n-dimensional subspace of B, (~) E, cE,+x (n = 1,2,...), (6) E =U,~xE, is dense in B.
19661
BANACH SPACES WHICH ADMIT l~ APPROXIMATIONS
197
Conversely, the following observation is due to S. M a z u r (cf. Bessaga [2]). PROPOSmON 6.1.1/f in Banach space B there exists a sequence of projections p oo ( . ) . = t satisfying conditions (~)-(tS), then B has a monotone basis.
Ile.ll
Proof. Define (e,)~=l inductively such that = 1 and eneE, N k e r P , - x for n = 1, 2, .... (For convenience, we set Po = 0.) Since the range o f P~-x is (n - 1)-dimensional, the kernel K e r P , - 1 has codimension = n - 1. Therefore the intersection E, ~ Ker P , _ i is non-empty. T o prove that ( e ,)~Qo= 1 is a m o n o t o n e basis for B, observe first that, since 11P, [[ = 1, e , + l e Ker Ps and e~ ~ E n for v = 1,2, ...,n.
t.e.ll ~_ IP~C~at, e.) [[ = I1~ 1 t.e.[ n+m
ll
n
for arbitrary scalars tl,t2,...,t,+ 1. Thus by induction II >--II L ~ t , e , II for arbitrary scalars t~, t2,-.-, t,+~ (n, m = 1,2,...). But the last inequality, together with (6), implies that ( e~ ) ,®= i is a m o n o t o n e basis for B. (c. f. [10])
Proof of Corollary 1.2.
This follows f r o m T h e o r e m 1.1, L e m m a 2.1 and
Proposition 6.1. It follows f r o m a result o f Lindenstrauss I'7] that, if B is a n~-space and if E is the range o f a projection of n o r m one f r o m B with dim E = n < + oo, then E is isometrically isomorphic to l~(*). Hence we can complete Corollary 1.2 as follows: COROLLARY 6.2. Let B together with a sequence of subspaces (E,) satisfy condition Oii) of Theorem 1.1. Then there exists in B a monotone basis (en)~=l such that (et, e2, ... , e,} spans E~ (n = 1, 2,...). Conversely, if (e~)~=t is a monotone basis for a rc~-space B, then B and the subspaces E n spanned by { e l , e z , . . . , e , } satisfy condition (iii) of Theorem 1.1. REFERENCES 1. C. Bessaga, Bases in certain spaces of continuousfunctions, Bull. Acid. Polon. SCi., Cl. III, 5 (1957), 11-14. 2. F. E. Browder and D.G. de Figueiredo, J-monotone non-linear operators in Bausch spaces, to appear. 3. M.M. Day, Normed linear spaces, New York, 1962. 4. N. Dunford and J. T. Schwartz, Linear Operators, I, New York, 1958. 5. D. G. de Figueiredo, Fixedpoints theoremfor non-linearGalerkin approximable operators, to appear. (*) Indeed, ifB is a n~-space, then, by Corollary 1 of [7, p. 66], B satisfies conditions (1)-(13) of Theorem 6.1 of [7, p. 62]. Since E is the range of a projection of norm one from B, the subspace E also satisfies the same conditions. In particular, E** is a ~l-space. Since E is finite-dimensional, E** = E. Thus E is isometrically isomorphic to In~.
198
E. MICHAEL AND A, PELCZYNSKI
6. V. I. Gursri~, Bases in spaces of continuous functions on compacta and some geometric problems, Izv. Akad. Nauk SSSR Sex. Mat. 30 0966), 289-306 (Russian). 7. J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 8. E. Michael and A. Pe~czy~ski, ,4 linear extension theorem, to appear in Illinois J. Math. 9. ~ , Peaked partition subspaces of C(X) to appear in Illinois J. Math. 10. V. N. Nikol'skii, Best approximation and basis in a Fr~chet space, Dokl. Akad.| Nauk SSSR 59 (1948), 639-642 (Russian). 11. A. Pelczyfiski, On simultaneous extension of continuousfunctions, Studia Math. 24 (1964), 285-304. 12. F. S. Vaher, On a basis in the space of continuous functions defined on a compactum, Dold. Akad. Nauk SSSR 101 (1955), 589-592 (Russian).
U
~
oF WASHINGTON,
AND UbriwP.srrY oF WARSAW, AND UNIVERSITYOF WASmNGZON
ON A CERTAIN BASIS INto BY
M. ZIPPIN* A B S I T ,A C T
A basis {Xn}~__1 is constructed in co such that there exists no bounded linear projection of co onto the subspace spanned by a certain subsequence oo ¢~o {X,~}t=1 of {Xn}n=1.
I. In~odu¢fion. A. Pelczyfiskiraised the following question ([3],Problem 4): Let {xn}n%I be a basis of a Banach space X. Is each subspace of X spanned by some subsequence {x.~}~°=l of {x.}~=l complemented in X? In this paper we show that the answer is negative by constructing a suitable example in co. Our main tools are the following two propositions: PROPOSmON 1. (See [1] Theorem 3.) l] +1 can be isometrically imbedded into l~" and every linear projection P of 12" onto l~+1 has norm
IIe I1 (. + 1)2-" [n/2
(In/2]
•
denotes the greatest integer ~_ n ]2.)
PROPOSmON 2. (See [2] p. 16, Corollary 3.) IrE is a finite dimensional subspace of a Banach space X for which X** is a P~ space and there exists a projection with norm c from X onto E, then E is a PT¢ space. (X is called a Py space if for every Banach space Z containing X there is a linear projection P from Z onto
x with IIP II If {x~}l~l is a s e t o f elements of a Banach space X then [x~]~,1 denotes the ~2~ the usual basis closed linear space spanned by {xi}i~1. We denote by tYets/--x 2n of 12" and by {f~}l = l the corresponding biorthogonal functionals in 12"= (l~')*. 2. Preliminary lemmas.
Denote by AI the matrix
:) Received Aug. 5, 1966 and in revised form Sept. 8, 1966. * This is part of the author's Ph.D. thesis prepared at the Hebrew University of Jerusalem under the suppervision of Professor A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful advice. 199
200
M. ZIPPIN
[Septembcx
and let A k (k > 1) be the matrix obtained from Ak_ 1 by substituting A1 for + 1 and - A~ for - 1. It is easily proved that 2-nAn is a 2 n x 2n symmetric orthogonal matrix. Denote by a~,~ the elements of An and 2 ~ by g(n). LV_~VtMA 1. A~ is obtained from A 1 by substituting + A k _ l f o r 1 and - A~_ t for - 1, k = 2, 3,4,....
The proof follows by induction from the definition of An. As a consequence of Lemma 1 we get LEmc~A2.
For 2 ~ n ,
l 1, g(n -- 1) < k < m < g(n) and every sequence of scalars
Z cl(e~ + e~+e(s_l) ) H II
i=l
g(n- 1)
~_
Z
t 11
c~(e'2+ e~+z(._ 1)) +
Z
B
i=g(n-1)+1
|=1
1 ,
1 < n < q g(m), it follows from (2) that
~ ciy~+l II~_
f=l
c,y,.÷~ U
i1~,-~ ,-, i=1
I1¢, ~., (=~q
~l
~
+1
|
Ci
1
Y~+I) II< 2 Ii~
ciym+lll
i=t
This concludes the proof of Lemma 6. 4. A non-complemented subspace of Co. Denote by {ei}i~ 1 the usual basis n Co and let U, be the natural linear isometry from l~(n)onto
~ai=g(.)
e "lg(n+ 1 ) - I n
= 0, 1, 2, ....
U.
cte7
= ~
\i=1
)
ciei+g(n)-I •
f=l
Put z ° = et and z~'= U.(y~) for n _>_1 and 1 n, q < g(n + 1) and r < g(m + 1) it follows from (3) that
204 (4)
M. ZIPPIN =o~=
c~z~
+ ,=1 ~ c,~+l-'+lz,
g(k) 2 max
~
k~n+l
2 max
~
1=1
{
max
k~_rn
Zcz
2
=0 \ i = 1
} ¢i~Zl~
~,=
c~z~
u/Ii ,
~., c~ ÷1z'÷1 ~1
ul
+ ~=1
The last inequalities show that the sequence {zT} in its natural order forms a basis in Co and Lcmma 7 is proved. By [1], p. 459, [xT]~ ~1(n) is isometrically isomorphic to /~+1 and since U, is a linear isometry, we get by Lemma 4 that [z~.]~ ,z(n) is also isometrically isommorphic to l~÷1. Suppose that P is a bounded linear projection from Co onto the subspace Y spanned by the sequence {z i }, ~i(~) n = 0,1,2, .... It is obvious that the sequence {z~}, ~ t(,) n = 1,2,... forms a basis in Z From the proof of Lemma 7 it follows that there exists a sequence of projections {Qn} from Y onto 1 then X* is an LI(/~) space for some measure/1. Here we shall prove that also the converse is true. In [3] Michael and Petczyfiski studied Banach spaces X which have the following property: For every e > 0 and every finite set A in X there is an integer n and an operator T : I ~ ~ X such that for every y E 1~ and such that the distance of x from T l~ is < e for every x s A. Here ! ~ denotes the space of all the n-tuples o f real numbers y = (21,22,..., 2n) with [] y [] = max, ]2i ]. These spaces were called in [3] a o~spaces. Since i : is a ~1 space for every n it follows easily that an a °° is an ,W~ space for every 2 > 1. Here we show that the class of a ~o spaces coincides with the class of the spaces which are ~ x for every 2 > 1. We consider only Banach spaces over the reals, but our result and its p r o o f are valid also in the complex case. We state now our main result. T , ~ O ~ M 1. L e t X be a Banach space. Then the following three statements are equivalent, (i) X * is isometric to the space Ll(Iz) f o r some measure IZ. (ii) X is an ,/V~ space f o r every ;t > 1. (iii) X is an a ~° space. F o r spaces X whose unit cell has at least one extreme point Theorem 1 can be also easily deduced from the results of [1]. The p r o o f of Theorem 1 presented
Received August 19, 1966. * The research of the second named author has been sponsored by the Air Force Otfice of scientific Research under Grant AF EOAR 66-18 through the European OffSet of Aerospace Research (OAR) United States Air Force. 205
206
A.J. LAZAR AND J. LINDENSTRAUSS
[September
here is, however, shorter than the arguments given in [1] from which the special case of Theorem 1 follows. A list of other properties equivalent to property (i) of Theorem 1 is given in [2, Theorem 6.1]. By combining the results of 1"3] with Theorem 1 we get immediately the following stronger version of Theorem 1 for separable spaces. THEOREM 2. Let X be a separable Banach space. Then the following two statements are equivalent. O) X* is isometric to the space Ll(lO for some measure g. (ii) X has a monotone basis {e~}i~l such that for every n the subspace of X spanned by {ei}~%1 is isometric to 1~ We pass to the proof of Theorem 1. As we have already remarked we need only to show that (i) ~ (iii). Let X satisfy (i) of Theorem 1, let A be a finite subset of X and let 0 < e < 1. In the definition of an a °° space it is clearly enough to consider sets A with It xl/-- 1 for every x e A (otherwise replace x e A by x/ll x II a n d ~ by e/maxx~a IIx ]l). so we assume that II~ II-- 1 for every x ~ A and let B be the subspace of X spanned by A. Let E o be the set of exposed points of the unit cell of B*. Let J~o be the set obtained from Eo by identifying every f with - f , and let ff be the quotient map ~b:Eo ~/~o. We metrize E o by putting d(dpf, dpg) = rain (llf- g II, Ilf+ Since B is finite-dimensional the metric space/~o is totally bounded. Hence, there is a finite number of subsets {Gl}~ffii of /~o such that G~n Gj = ~ for i ~ j , E o = 1,.J7=1G~ and Gl has for every i a non empty interior and a diameter < e. Since e < 1 there is for every i a subset G~ of Eo such that c~- l Gi = Gi U - G,, G, n - G, = ~ and < ~ f o r every f, g e G,. For every i pick an f~ E G~ such that fffi is an interior point of G~ and let x~ e B be such that A(x,)= II x, II = IIf, II = ~ and f ( x , ) < 1 for every f ~f~ in B* with
gll).
IIf-gll
Ilfl[ = Let E = [,.J~=1 Gt and let l °°(E) be the Banach space of all real-valued bounded functions on E with the sup norm. Let the operator U:B--. l°°(E) be defined by U b ( f ) = f ( b ) , b e B, f e E. Since the unit cell of B* is the closed convex hull of E u - E we get that U is an isometry. From our choice of the x~ and f~ it follows that there is a c5 > 0 such that If(x,) I < 1- a for every i and every f ~ E ~ G~. We assume as we may that fi < min (2]3, 1 - t). Let y~ e l°°(E), 1 < i < n, be defined by Yt(f) = 1 if f e Gl and Yl(f) = 0 if f e E ~ G,. By our choice of fi we get that 11--< ~-Xn fact, if f ~ E N G~ then
II -Wx,-y,
] ~ - ~ U x , ( f ) _ y,(f)] =
la-~f(x,)l
_< ~ - 1
--
1,
while for f ~ G~ we get (since ~-~f(x~) ~ (1 - ~)/~ ~_ 1)
Ia-lUx,(f)-
Y,(f)I = I~-' f (x,) - 11~ ~-~ - 1.
1966]
BANACH SPACES WHOSE DUALS ARE L~ SPACES
207
Since X* is an LI space there is (see e.g. [2, Theorem 6.1 (3)]) an operator T from l~(E) into X whose restriction to UB is equal to U-1 and with norm II T II < (1 - ~ + ~/2)/(1 - ~) We have, in particular, that for every 1 < i < n
Ill-'x,- Ty, ll =
II~-ITUx, -
Ty,[I---IlTllll~-'ux,-y,l[
-- 1 - ~ / 2 Let Y be the subspace of I~(E) spanned by (Yt}~'=l. Clearly, Y is isometric to 1~ We claim that for every y e Y
(1 - 2n)II y
II
1 - 2~ In order to conclude the proof that X is an a ~ space it is now enough to show that for every x e B with fix I!--1 there is a y e Y with II T y - x II < 2~ Take y = ~7=~f~(x)y, E Y. Then I1Y - Ux II < ~ (recall that the diameter of each G, is < ~) and hence
11r y -
xll < 11TII H Y - U~II z~(1 + 8 ) < 2 ~
and this concludes the proof. REFERENCES 1. A.J. Lazar, Spaces of affine continuous functions on simplexes (to appear). 2. J. Lindenstrauss, Extension of compact operators, Mere. Amer. Math. Soe. 48 (1964), 3. E. Michael and A. Pelezy~ski, Separable Banach spaces which admit In~ approximations, Israel J. Math. 4 (1966), 189-198. THB HEBP~W UNIVERSITY OF JERUSALEM
RECURRENCE OF SUMS OF MULTIPLE MARKOV SEQUENCES BY
P. HOLGATE ABSTRACT
A d-dimensional random walk on a lattice is studied in which each step is bounded, and may depend on the previous m steps. It is proved that if trivial cases are excluded, there are no recurrent points for d _~ 3, and conditions are given for the existence of sets, recurrent conditional on the first ra steps, for d = 1, 2. Let X t , X 2 , " - , be a sequence o f d-dimensional random vectors with integer valued components, and let S , , = X 1 + ...+X,,. In the case where the X~ are mutually independent, each taking each of the values + e~ with probability 1/(2d), where ej is the vector with unity in itsj-th component and zeros elsewhere, P61ya proved in [8] that Pr{Sn = 0 for an infinity of values of n} (la)
=
1, if d = 1,2,
(lb)
= 0, if d > 3.
The recurrence problem when the Xi form a Markov chain has been studied in generality only by Gillis [5], although other writers have dealt with the distribution of Sn in this case for d = 1, and Seth [9] has obtained some further results on recurrence for this value of d. Gillis proved that P61ya's result still holds if the Xi have the same range as above and satisfy symmetry conditions of the form (2)
Pr{Xn = ¢ [ Xn-I = ¢'} = Pr{X, = - ¢[ X , _ l = - ¢'},
and simplifying restrictions which he conjectured to be inessential, although his p r o o f o f (lb) did not cover the odd integers ~ 3. In this note I suppose that the Xi form a multiple Markov chain o f arbitrary dependence, and are bounded, and by means of a different method based on the recurrence properties o f finite Markov chains and generalisations of P61ya's result, [1,4], give a necessary and sufficient condition for (1) to hold essentially. (Complete enumeration o f the particular cases corresponding to degeneracies in the transition structure would be tedious). However, unlike Gillis's analysis of the Received June 9, 1966 and in revised form Sept 26, 1966. 208
R.F.CURRENCE OF SUMS OF MULTIPLE MARKOV SEQUENCES
209
backward equations of the Sn process, this method does not readily provide information on the distribution of Sn. Let us suppose that the X~ form a multiple Markov sequence of order m, [2, pp. 89, 185], subject to the condition that if X[ j)is the jth component ofXtthen with probability one d
(3)
IIx, II =
I "1
B.
Let Z~= (Z~1), ..., Z~m)) = (Xi,... , Xi-m +i)be defined for i ~ m as the m dimensional vector whose components are consecutive vectors of the sequence {Xt}. The sequence {Z~} is a simple Marker chain M, [2], and by (3) it has a finite state space C. To avoid the uninteresting complications referred to above it will be assumed that M consists of a single ergodic class, (assumption E). Now choose an arbitrary state ~oeC, and denote by 7"1,T2, ..., the random values of n for which Z,,= ~o and define the following random variables, k
U|= T[+I- Ti, Vi=ST,+I--ST,, Wk= ~a V[ i=l
(4)
Xt(~)= 1 if Zt = [, = 0 otherwise,
Tl+1 v,(O = ~ x,(O. /=TI+I
It is known from recurrence theory, [3, Chapter 15, in particular Exercises 19,25] that the Ul are independent, and for i =>2 are identically distributed with Pr{U i = k} = a t where for some positive A < 1, ak < 2~. The V, are also independent and for i > 2 are identically distributed. (This approach and similar notation has been used by Katz and Thomasian [7] to obtain probability bounds for sums of functions defined on Marker chains.) Further, since M has a finite state space it admits a stationary distribution p([) with the property that for some constant A, #v,(O = Aft(O, ( = v(O). Let ~(~) be the marginal distribution induced by fl on the first component of ~.
If conditions (3) and (E) hold, then for the sequence V~ defined above, (i)
,V~=A
Y~ ~ ( 0 ,
ill v, II < oo.
210
P. HOLGATE
[September
Proof. (i) TI+t
ce, =
~
~*Xj
j=Tl+l
=
2
t;°)v(O
=
A ~ ~")P(O
=
A
~(0.
E
oO
(ii)
llv, ll _
0, Pr{S, = a, Z~ = ~ [Zm = ¢'} > 0. Trmo~M 1. Under assumptions (3) and (E), for d = 1, 2; (i) if ?E~(¢)= O, every point a that is possible given ~' is also recurrent given ~', (ii) if X ~ ( ¢ ) # O, no point is recurrent. Proof. (i) Consider the sequence Wk defined in (4), taking ~o = ~'. By Lemma 1, (i) it satisfies the conditions for the existence of a recurrent set, given in 1"1,4]. Let us choose a point p belonging to this set and suppose that a is a point, possible for S, given ~', not belonging to it. Then Pr{St = e l i ' ; Qt < l < O,+a} > 0 for some i. But by (5) this holds for all i, and hence a is recurrent. The set of possible values given ~' will in general consist of a subset of the cosets of a subgroup of the additive group of one or two dimensional integers respectively.
1966]
RECURRENCE OF SUMS OF MULTIPLE MARKOV SEQUENCES
211
(ii) The points which are possible for Wk given ( ' are transient for Wk given ('. Let p be such a point. Now suppose that some point a is recurrent for S.. Then by the finiteness of C, some pair (a, (") must be recurrent. Let T[, T~,..., be the values of n for which S. = a, Z, = (". Now Wk=P for some k if and only if Sl = p, Zl = ( ' for some I. Since p is possible for Wk we have Pr{S, = p,Z, = ( ' l ~ ' ; T ( < l < T[+,] > 0 for some i, and hence for all i since (S, = tr, Z,, = [") is a regenerative event in the sense of (5). This implies the recurrence of p and the contradiction establishes the result. If the symmetry condition Pr{X.
= ¢.1x._1
Pr(X.
=
= ¢._....,x._.
= ¢._.}
=
(6) -
=
-
¢.-1,-..,x.-.
=
-
n > m,
is imposed on the transition probabilities, it is easy to see that the stationary distribution on m must be symmetric, f l ( [ ) = / 3 ( - O, hence ~(~)= c t ( - ~) and condition (i) of Theorem 1 is satisfied. It can readily be seen that (6) is satisfied by the correlated random walk studied by Gillis in [5], mentioned earlier, and also by the two dimensional process which he discussed in [6]. In both these eases all points are recurrent whatever the initial step. TrlEOREM 2. For d >=3, every point is a transient point of S,. Proof. Suppose there exists a point a such that Pr (S. = a for an infinity of n) = n > 0. Then since the state space C is finite there must exist a ~o such that S. = a , Z , = (o for an infinity of n, with probability re. Using this [o, define a sequence V~as in (4). Then we have a sequence of independent random variables, for d => 3, for which it is not true that every point is transient, which contradicts the assertions of [1, 4]. I am most grateful to the referee for pointing out an error in my discussion o f Lemma 1, (i), and showing that Theorem 1 held under wider conditions than I had originally imposed. REFERENCES
1. K. L. Chung and W. H. J. Fuchs, On the distribution of values of sums of random variables, Mere. Amcr. Math. Soc. 6 (1951), 1-12. 2. J.L. Doob, Stochastic Processes, Wiley, New York, 1953. 3. W. Feller, Probability Theory and its Applications, I (2rid Ed.), Wiley, New York, 1957. 4. F. G. Foster and I. J. Good, On a generalisation of Pdlya's random walk theorem, Quart. J. Math. (2) 4 (1953), 120-126. 5. J. Gillis, Correlated random walk, Proc. Camb. Phil. So¢. 51 (1955), 639-651.
212
p. HOLGATB
6. L Gillis, A random walk problem, Proc. Camb. Phil. Soc. 56 (1960), 390-392. 7. M.L. Katz and A. J. Thomasian, An exponential bound for functions of a Markov chain, Ann. Math. Statist. 31 (1960), 470-474. 8. G. P61ya, ~)ber eine Aufgabe der Wahrseheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz, Math. Ann. 84 (1921), 149-160. 9. A. Seth, The correlated unrestricted random walk, J. Roy. Statist. Soc. B. 25 (1963), 3 ~ A.O0.
]BIOld]~rRIC~SECTION, T ~ NA~tnte CONSnVANCY, LONDON
APOLOGY The Editors apologize to the author of the article: "On the mean length of the chords of a closed curve", which appeared in Vol. 4, No. l, for printing his name incorrectly. The author's name should read G~bor LOK(~.