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. Similarly we see that is an entire function of order p = oc/(oc - 1 ) and type (oc - 1 )oc-a/(rx- 1>. The determinations of the order and type of the entire functions and were carried out for the functions associated with the densities Prxy ( x), that is for the case A == 1 . Similar results can also be obtained if A # 1 . Expansions analogous to (5 . 8. 8a) and (5. 8.8b) can be derived easily ; let a�> (A) and b�> (A) be the coefficients in these series. It follows from (5 .8.2b) that a�> (;t) = a�> Ak, while b�> (;t) = b�> A- k/rx. We conclude from (5 .8. 1 2a) and (5 .8. 1 2b) that the order of the entire functions associated with oc, y, A) is given by p = ( 1 - oc) -1 if 0 < oc < 1 but by p == oc(oc - 1 ) - 1 if 1 < oc � 2. The type 1: = AP (1 - oc)rx/ ( 1 -cx> for 0 < oc < 1 , but 1: = A -p/ct (oc - 1 )oc -rx/(ct-1> in the case where 1 < oc � 2. It follows easily that the function is also an entire function of order p = rx(oc - 1 ) - 1 and type 7: = A - p/rx (oc - 1 )oc-ct/(ct- n . We still have to consider the case ex = 1 . Since the representation (5 .7.2 1 ) is not valid for oc = 1 we must use the canonical form (5 .7. 1 9) as our starting-point. If fJ = 0 we have f = exp I I) ; this is the characteristic function of the Cauchy distribution, and the corresponding frequency function is
2 (z)
'Y(z)
; (z)
'Y(z)
p(x;
'F(z)
( t)
(-c t
p(x; 1 , 0, c) = (x 2c+ c 2) . n
x
ic
This is a rational function with poles at the points = + and is therefore regular for all real The radius of convergence of the Taylor series of 1 , 0, around the point = 0 is equal to We study next the case fJ # 0 in view of the fact that a relation similar to (5 .8.2c) is also valid for oc == 1 , it is no restriction to assume that fJ > 0 . Using the reasoning which yielded the expression (5 .8. 1 b), we see that
p(x;
x.
c)
(5 .8. 1 3 ) We write
z
x;
c.
p(x; 1 , {J, c) = ! Re J : exp { - itx - c{1 + � i log t] } t . g(z) == - ixz - cz - 2(J ciz log z, a
1t
where is a complex variable, and consider again the closed contour r used in deriving (5 .8.8a) . As in the earlier discussion we show that lim
(5 .8. 1 4a)
T---+0
f
Or
exp
[g(z)] dz = 0.
(•> The use of the notation p(x ; 1 , 0, c) cannot create any confusion since the symbol f'(x ; oc , y, ;'\), i n troduced on page 1 38 and based on the representation (5.7. 2b ) , i s not dt�fi ned for
« =
1.
146
CHARACTERISTIC FUNCTIONS
� lim I exp [g(z)] dz = 0 .
Subdividing the range 0 � cp n of integration along small cp 0 , one can also show that (5 .8. 1 4b)
R-+ ro
C
R
at a sufficiently
On
The assumption that fJ > 0 is needed in deriving (5 .8. 1 4b ). One con cludes finally from Cauchy ' s theorem, (5 . 8 . 1 4a) , (5 . 8 . 1 4b) and (5 .8. 1 3 ) that (5 .8. 1 5 )
p(x; 1 , {3, c) = ! I: [si
n (1
+ [J)ct] exp
p(x;
{ - xt - � ct log t} dt.
We wish to study the analytic character of 1 , {J, c) in the case where fJ # 0 . Without loss of generality we can put c = 1 ; for the sake of brevity we write instead of 1 , {J, 1 ) . We expand e - xt in (5 . 8 . 15) and ex change the order of summation and integration. In this way we obtain {1 t log t t 1 a:J < 1 ) k 2 a:J f' [ sin ( 1 ) t p :x!' P ex = - "\:'
P,a (x)
or
pp(x)
n6
p(x; I
_
k!
+ ]
o
(
d )
n
(5 . 8. 1 6 ) where (5 .8. 1 6a) Let n1 = (n/2f3)rJ, where 'YJ > 1 may be chosen arbitrarily large, and put t1 = exp (rJ1). We write the integral in (5 .8. 1 6a) as the sum of three inte grals ]1, J 2 and ] 3, taken over the intervals (0, 1 ), (1 , t 1) and ( t1 , oo ) res pectively, and estimate ] 1 , ]2 and ] 3 • Since max [ - t log t = e - 1 \Ve see that
]
(5 .8. 1 7a) and (5 .8. 1 7b )
11 2 1
tk + 1
� (k � 1 )
(the estim ate for J 2 follows from the fact that t log t We have
so that I Ja l
>
� I� t e ( - ?� t log t) dt. ''
x
p
0 for t
>
1 ).
INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS
t1 , we get ro ro k 1 1 3 1 � f t e YJt dt � f tk e -tJt dt; t l therefore (5 .8. 1 7c) I f 3 1 � 'YJ - k - l k ! It follows from (4. 8 . 1 7a), (5 . 8. 1 7b ) , (5 .8. 1 7c) and (5. 8. 1 6a) that k 1 [ k k +1 'I'J 'I'J C t 1 k l ak l ::::; TJ - - (k + l ) ! + (� + l ) ! + l J
1 47
In view of our choice of
0
+
or (5 .8. 1 8) We see therefore that
ak l 117c � '1'}- 1 [1 + o( l)] . Since 'YJ can be arbitrarily large, we conclude that lim sup I ak 1 1 /lc = 0 I
so that pp ( x) is an entire function. We rewrite (5 .8. 1 8 ) in the form 1 (5 .8. 1 9) I I = Ok 'YJ -Tc- [1 where Ole is a real number such that 0 � O � 1 . Since PtJ (x) cannot be a polynomial, there exists necessarily a subsequence okj of the o k such that O�c > 0 . In order to simplify the notation we write in the following 0 1 1 instead of 01ci · Using (5 .8. 1 9), we see that j log j j log j = O(logJ). log I (j + 1 ) log '1'} - log hence log hm sup og - 1 = oo. k-+ oo 'fherefore pfJ (x) is [see (5 . 8 . 1 2a)] an entire function of infinite order. We summarize these results in the following statements.
ak
+ o(1 )]
a1 1- 1
k
O; + o( l )
-
.
k k I I ak I
frequencyfunction of a stable distribution with character eorem 5 . 8 .5 . The1 has the form istic exponent 1 1 (x -ct) for x 0 nx = A) 1 p( x ; ( I x l-ex ) for x < 0, 2 nx 'J'h
rx
rx, y,
�-
148 where 1 ( ) and/(1 ) 2 (z) are entire functions of order (1 -ex) - 1 and type 1: = AP ( 1 -ex )r1. -r1. . The frequency function of a stable-distribution with exponent 1 is an entire function of order ex(ex- 1) 1 and type 1: = A-p/\1. (ex- 1 ) �-\1./(cx - 1) . Theorem 5 .8.6. Stable densities with exponent ex 1 are entire functions of infinite order 'if {3 0 but are rational functions if {3 = 0. In this case they have poles at the points ic and - ic. CHARACTERISTIC FUNCTIONS
z
p =
oc
p ==
>
==
#
5. 9
Asymptotic expansions and integral representations of stable densities It is sometimes convenient to have asymptotic expansions of stable density functions. In Section we will also use a representation of the derivative of a stable density by an integral. In the present section we derive some of these formulae. As a first example we derive an asymptotic formula for stable densities with exponent < < which is valid for large positive values of and We see from that
5.10
1 ex 2, Pcxx. y (x), (5.8.1b) (5.8.7) Par (x) ! Re {exp (i;;) J � exp [ - ixu exp (i;;) - ua] a } We introduce a new variable by putting u tx-1 e-in/ o ;oc (oc + y - 1) and where I () I Ik = f � t"'k exp ( - t eicl>o) dt. We then obtain from (5. 9 .1) the expression n . 1 1 ) in(y { ] mk Tc . C1. k /2 Jk = x (x exp e (1) (5.9 . 2 ) Pcxy ) -nx Re [ 2ex � k! n
1)
=
�
=
1
k=o
1)
'
149
INFINI TELY DIVISIBLE CHARACTERI STI C FUNCTIONS
In order to determine the integral Ik we change the path of integration from the positive real axis to the line == u e-icf>o , where 0 � u < oo . To show that this is permissible we consider the function g(z) == zak exp z eicf>o and the circular arc r lim f'-'}- 00
{z : z ==
r
t [- ] eicf> } , - cp0 � cp � 0, and conclude that
f g(z) dz = 0 and also lim J g(z) dz = 0 . In this way we see that r
=
r-�
l1c = exp
r
[-icp0rxk-icp0] r(rxk+ 1 ).
We substitute this into (5 .9.2) and obtain the asymptotic formula (5 .9.3) 1) 1 n (-1 r( ( sin ) = x -ak + O ( - a (n + ) ) 2: a, 2
l k [ ) k rxk + n l ) (rx x x +y P J k! JlX k= O as x and 1 < rx 2. vVe compare formula (5 .9.3) 'vith (5 .8.6) and see that the series in (5 .8.6 ) is convergent for 0 rx < 1 but is still useful as an asymptotic series if 1 < rx 2. It is sometimes of interest to have asymptotic expansions as x tends to zero. We treat as an example the case where 0 < rx < 1 while x 0. In -
0,
� ,x2/(a - Il [ b(cf) exp [ - xa./(a. - u a(c/>)] dcfo sin rxcp ) 1 /(l -a. ) sin ( 1 - rx )� _ ( where a(cp) - Sln cp .Sln rxcp and '2/ ( 1 - rx> 2rx cos cp sin (1 - rx) � ) sin rx cf> ( ( ) 1 = b (cp) (1 ) sin c/J sin cp while P�a. (x) 0 for x < 0 . The last statement follows from theorem 5 .8.3 , so that we have only to
(5 . 9 . 7)
P�a (x) = •
_
- IX
IX
=
prove (5 .9.7).
(life) We write here and
to the variable
x.
i n the
followin g Prxy (x) for the derivative of
Pa.y
(x) w i th respect
INFINITELY DIVISIBLE CHARACTERISTIC FUNCTI ONS
151
We differentiate (5 .8. 1 a) to get an expression for p�, (x) which is similar to (5 .8. 1 b) and see that P�r (x) =
(5 .9. 8)
oo ( - it) exp [ - itx - ta. e- inyf2] dt,
�n Re Jf o
provided that rx # 1 . We introduce the new variable t = vx1/ [n sin � + n" cos r�.(� + �)] } d�. ..
We select a cp0 such that
(
. n n( l - ex) 0 < 1> o < min 2 ' 2cx
)
153
INFINITELY DIVISIBLE CHARACTERISTI C FUNCTIONS
and conclude that
- nxoc/ (rx- l> sin cp
1 2/ (oc - l> n 2 n / 2 x exp
J
+ n-
This means that
cb o
J
Qn
h (z) dz =
o( l )
dcp . as n
--+ oo ,
so that (5 .9. 1 1 b) is proved. We finally note that therefore (5 .9. 1 1 c)
Re
J
�r
g( - iy) is real for real
y;
h (z) dz = 0.
It follows from (5 .9. 10), (5 .9. 1 1 a), (5 .9. 1 1 b) and (5 .9 . 1 1 c) that 00 h(z) dz = Re h(z) dz. P � oc (x) = Re ( 5 .9. 1 2)
J
f
0
rn
In view of the definition of the contour r we know that ei� cp)] = Re ei� cp)], so that
g[ p(
g[ p(
(5 .9. 1 3 ) We substitute into (5 .9. 1 3) for p(cp) the expression given in (5 .9.9) and see, after an elementary computation, that ( 5 . 9 . 1 4)
g[e��. p( 4>)] = =
a(cp)
1 1 a ( 1 ) sin sin IX (cf + �) sin
(cfo+ ;)
- a (cf + ;)
[(1 - IX)(cp + �)J sin IX ( cfo+ ;)
is defined in the statement of theorem 5 .9. 1 . Since where the function z == cp) eicf> on the contour r we see easily that cos 2cfo] drp sin 2cp Re (iz dz) = and obtain from (5 .9 .9) the expression
p(
where
-[pp' + p 2 Re (iz dz) = [p(cf ) PB (cf + ;)acp ,
OS_ll(�_ sin cp -_2 cos cp sin rup ( ��� cp) = cos cp cos zcp. B (l - ex) stn ocrp
+
1 54
CHARACTERISTIC FUNCTIONS
It follows from (5 .9 . 1 2) and (5 .9. 1 4) that (5 .9. 15) P�� (x) =
! x2/(a- 1) J : [p (� - ;)J 2 B(�) exp { - x�/(o:- l> a(cp) } dcp .
A simple computation yields the expression 2oc s � i� ( 1 oc)� B(�) = (5 .9 . 1 6) - 1. s1n We put (5 .9 . 17) si� n oc cos � si� ( 1 2 > _ -cx ;n 2 B(� ) = = ( 1 - oc) s1n s1n 2 and obtain the statement of theorem 5.9. 1 . The following alternative expression for B( ) is easily obtained from (5 .9 . 1 6) : 1 si� (2 = B(� ) ( 5 .9 . 1 6a) -1 1 - rx s1n and this will also be used.
�� ; f - rx rx
[ rxcpcfo]
2 [ ) b(cp) p(c? J cp
{rx
rxcfo
{
rxcp
rx)cp - t}
oc)cp }
Corollary 1 to theorem 5 .9 . 1 . The function a(cp ), defined in theorem 5 .9 . 1 , is strictly increasing in the interval [0, n] . Let, for cp fixed, VJ (oc) rx cot rxcp - cot �- Clearly '!fJ(1 ) 0 while �: 0 , =
=
so that
rx cot rxcp
cp
cot for 0 < < 1 . Moreover, it is easily seen that 1 cot cot + (1 cot (1 log a(� ) = d� 1_ > (1 cot ( 1 - cot > 0, which proves the corollary.
rx d
>
�
2 2 cp rx) {rx rxcprx rx)cp cp rx)
rx)cp }
Corollary 2 to theorem 5 .9 . 1 . The function b(cp), in the statement of theorem 5 . 9 . 1 , has exactly one change of sign in the interval [0, n] . In view of (5 .9 . 1 7 ) and (5 .9. 1 6a) it is sufficient to show that u(�) oc si�s1n(2rxcp- rx)cfo _ 1 =
has exactly one change of sign in [0, n] . We note that u(O) = 1 - fX while u(n) == - ( 1 + oc)/(1 - ex) < 0, so that at least one change of sign occurs in
155
INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS
the interval. An elementary computation shows that
u ' (c/> ) where v(cp) so that
v '(cp)
==
=
=
v
(cf> ) rxcf> 2 sin (:/..,
(1 - rx) sin 2cp - sin 2(1 - rx)cp
-
2( 1 - (:/..,) [cos 24> cos 2 ( 1 - (:/..,)cpJ.
It is then easily seen that there exists a unique value cp0 such that
n,
0 < cfoo < while v' (cp 0) 0 , and we get cp 0 n /(2 - rx). It follows that v(cp) has exactly one minimum inside [0 , n] , so that v (cp) and therefore also u ( cfo has at most one change of sign in this interval. The statement of the corollary follows immediately from (5 .9. 17). It is also necessary to derive a result similar to theorem 5. 9 . 1 for the cas e where the exponent rx > 1 . ==
=
)
Theorern 5 .9 .2. Let 1 < a. 2 ; then for x 0, 1 (5 .9. 1 8) P�.ll.- 2 (x) n x2/(ll.- 1 ) J :n::n:/ll. b1 (0) exp {xa/ tll.- 1) a1 (0)} dO, while for x < 0, 1 (x) (5 .9. 1 8a) P�.ll._ 2 n-- l x \ 2/(ll.- 1) J b2 (0) exp {-lx jll./(a- 1) a2 (0)}d0, where [ [ sin () 1 / (ll. - 1 ) sin ( rx - 1 )0 - sin ()] 1 / (cx - 1 > sin ( rx - 1 ) fJ . , a 2 (fJ) a1 ( fJ ) stn rxO s1n rx()J s1n rx() s1n rxfJ and [ - sin ()] 2/ (ll. - 1) { 2rx cos fJ sin ( rx - 1 ) 0 1 b 1 (0) } (a. - 1 ) sin rxO sin rxO [ sin () 2 / (cx - 1 > { 2cx. cos () sin ( - 1 )0 } 1 b 2 (0) sin rxOJ ( rx - 1 ) sin O I n order to simplify the notation we write p'(x) instead of P�.cx - 2 (x) in the proof of theorem 5 .9.2. We consider first the case x 0 and substitute cx - 2 into (5 .9.8b) and see that oo p'(x) n� x2/ l<X- l) Re J (iz) exp {x"/l<X- l ) g(z)}dz
-
=
o :n:jct.
=
==
•
•
.
.
_
=
oc
==
_
•
rx
>
?'
==
=
where
o
156
CHARACTERISTI C FUNCTIONS
We.,..determine first, as in the proof of theorem 5 .9 . 1 , a curve along which g(z) is real. It is easily seen that the curve r 1 , given by 1 / n n n - cos � . . = p et4> w1th p = p(�) = z n . rx - -2 � � � 2 s1n rx � + 2
(
)
( ) satisfies this requirement. Moreover, p (� ) = 0 while p (: - ;) = oo. Let
Zn = n eic/> n be the point of intersection of r 1 and the circle of radius n with centre at the origin and write Cn for the arc of this circle which is located in the first quadrant between the real axis and the point Zn It is not difficult to show that •
f
c,.
\ o(l)
(iz) exp {x"f exp [xcx/ (cx - 1 > g ( p eicl>)] d� = 0 . + nj n
cx - /2 We use the fact that g [p(� ) ei4>] is real and introduce the new variable
() = 4> +
� and
obtain after a somewhat tedious but quite elementary
computation formula (5 .9. 1 8). We consider next the case x < 0 and see from (5 .8.2c) that P : .y (x ) = - P � . - y x I ) (x = - I x I < 0). We put y = rx 2 and write again p ' (x) for P�,(f. - 2 (x) and see from (5 .9.8a ) that 1 (zz) exp { - 1 x l cx/ ((1. - t > g (z) } dz p'(x) = x 1 2 / (cx - l ) Re (5 .9 . 1 9)
(I
J oo
-I
o
n
where
•
g (z) = iz - zcx eincx/2 • Again it is easily seen that lm g(p eicf>) = 0 if z defined by p(cp )
=
sin (4> + �) sin � (4> �)
1/(CX - 1 )
+
=
(-2 � n
p ei4> is on the curve r 2
¢ �
( �) (:Y
so that g(z) is real along r2• We have p -
=
;
n be the point of intersection of r and the circle of radius n with so that
oo
=
==
Zn
=
2
centre at the origin, and let Cn be the arc of this circle located in the first quadrant between the real axis and the point Zn . Then
f On (iz) exp {- I x la./(a.- 1> g(z)} dz o( 1 ) (n =
� oo
).
- ioc- 1/(a.- 1>]
We consider the contour which consists of the segment [0, of the imaginary axis, the arc of r 2 between the origin and Zn , the arc Cn and the segment [n, 0 of the real axis. We apply Cauchy ' s theorem and let tend to infinity and conclude that
n
]
Re
{ f (iz) r
,
exp [ -
I x 1 "'1("' - 11 g(z)] dz + J: (iz) exp [ - 1 x 1"'/(<X-ll g(z)] dz} =
It follows that
p ' (x)
=
! 1 x 1 2/!<X-ll
n
·
0.
J (iz) exp [ - 1 x la/(a.- 1>g(z)] dz. r.
Using the argument which we employed before, we obtain (5 .9. 1 8a).
Corollary 1 to theorem 5 .9.2. Let
X > 0.
1 < rx < 2 ;
b1
then P�.a.- 2 (x) < 0 for
To prove the lemma we need only show that (0) < 0 for n/rx < (} < Since sin rxO < 0 in this interval it is sufficient to show that (O) = (rx - 1 ) sin rx0 - 2rx cos 0 sin (rx - 1 )0 > 0 for nj rx < 0 < n . A simple computation shows that (O) = rx sin (2 - rx)O - sin rx0 0.
n.
y
y > 2 to theorem 5 .9.2. The function a 2 (0) which occurs in (5 .9. 1 8a) is Corollary strictly increasing in the interval [0, / ] Corollary 3 to theorem 5 .9.2. The function b 2 (0) which occurs in (5 .9. 1 8a) has exactly one change of sign in the interval [0, / ] n oc .
n rx .
'r'he proof of the last two corollaries is analogous to the proof of corol laries 1 and 2 to theorem 5 .9. 1 . 'fheorems 5 .9. 1 and 5 .9.2 expressed the derivatives of stable frequency functions in terms of certain integrals. V. M. Zolotarev ( 1 964) derived somewhat similar representations for stable distribution functions. In the
1 58
CHARACTERISTIC FUNCTIONS
next theorem we present his result ; we use here the notation of formula ( for stable characteristic functions. The characteristic function of a stable distribution with exponent r:t.. =1= 1 and parameters = A = 1 , y = K( r:t.. ) d is then given by . nK (r:t..) z log f(t ; r:t.. , d) = - I IC( exp •
5.7.23)
{ t
t
-
0 then
We write
#
F (x ; r:t.. , d) =
where
f
1 n/2 �(1 - d) + exp { - Vex (x, cp) } dcp, r5j2 n n 1 n/ 2 exp { - Vex (x , cp) } dcp, 1-1WK ) TC - (cx /2cx
V (x, cp) = xcx/ (ll. - l) C( If r:t.. =1= 1 but x =
f
ij r:t.. < 1 ij r:t.. > 1
1 / ! <X <X l [( - 1 )cf> + � cos cp one has r:t.. , d) = ![ 1 - dK (r:t..) /r:t..] . In the case where x < one obtains the corresponding representations from the above formulae and the relation 1 - F ( - x ; r:t.. , d) = F(x ; r:t.. , - d ). For the proof we refer the reader to Zolotarev ' s paper, quoted above. This paper also contains a similar formula for r:t.. = 1 .
0
•
F(O; 0
5.10 Unimodality of stable distributions In this section we prove the following theorem which is due to I. A. Ibragimov-K. E. Czernin.
Theorem 5 . 10. 1 . All stable distributions are unimodal. The proof is carried out in several steps. We first study symmetric stable distributions (i.e. y = 0) ; then asymmetric stable distributions with ex treme values of the parameter y [i.e., I y I K (r:t.. ) or equivalently I d I = 1 , =
respectively I fJ I = 1] ; and finally arbitrary stable distributions.
Lemma 5 . 10. 1 . All symmetric stable distributions are unimodal. Theorem 4.5. 3 states that the function 1 f(t) + I t ,.. (0 < � 2) =
1
IX
1 59
INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS
is the characteristic function of a symmetric unimodal distribution. According to theorem 4.5 .5 this is also true for the function 1
We conclude finally from theorem 4.5 .4 that = exp = lim n
f(t)
n� oo
(- I t let)
f (t)
is also the characteristic function of a symmetric unimodal distribution, so that the lemma is proved.
Lemma 5 . 10.2. Stable distributions with unimodal.
Iy I
=
K (rx)
and rx
=1= 1
are
b(cp
We consider first the case rx < 1 . The function ), which occurs in the statement of theorem 5 .9. 1 , has, according to corollary 2 to theorem 5 .9 . 1 , exactly one zero in the interval (0, n) . Let a be this zero and write formula (5 .9.7) in the form p:m (x)
= ! x21
n
)
=
f(t 1 1 , c1 , C2 ) ·
Since we have already shown that stable distributions with exponent rx # 1 are unimodal, we conclude from (5 . 10.5) and from theorem 4.4.4 that stable distributions with exponent rx = 1 are also unimodal, so that theorem 5. 10. 1 is completely proved. 5. 1 1
Self-decomposable distributions
We defined stable distributions by means of the functional equation ( 5 . 7 .2). It is possible to introduce other classes of characteristic functions in a somewhat similar manner. As an example we mention the characteristic functions f(t) which obey the relation (5 . 1 1 . 1 ) f ( t) = f ( ct) fc ( t)
for every c (0 < c < 1 ), where fc (t) is some characteristic function. The functions (5 . 1 1 . 1 ) were introduced by P. Levy and A. Ya. Khinchine [see Levy (1 937a), p. 1 92, or second edition (1954), p . 1 95 ] , and one calls this family of characteristic functions the class of self-decomposable
1 62
CHARACTERISTIC FUNCTIONS
characteristic functions [Loeve ( 1 955)] . In the present section we dis cuss some properties of this class.
Theorem 5 . 1 1 . 1 . All self-decomposable characteristic functions are infinitely divisible.
We first show that a function which satisfies (5. 1 1 . 1 ) never vanishes. We give an indirect proof and assume that f(t) has zeros. Then there exists a It follows from (5 . 1 1 . 1 ) such that f ( ) = O while f ( ) # O fo r l l < = 0 while fc # 0 for I I < that fc We see from theorem 4. 1 .2, putting = 1 and = 2, that
t0
(t0)n
t0
(t) t t0/
[
t
(�o) 2]
t t0 • t t0 •
l fc (t o) 1 2 1 . f( t o/2 ) . . . . . . t o b c / 1s contmuous m c, we ob tam a contrad1ct1on S mce J c (2) ' f( t o Z ) choosing c sufficiently close to 1 , so that the functions f(t) an d fc ( t) never 4 1 - fc
.f
� 1-
=
y
=
vanish. To prove the theorem we note that
and see that (5 . 1 1 .2)
t
n
(t). (t) J J k. n k= l IT
=
Since f ( ) is continuous and never vanishes, we conclude that lim fk.n
k
k n)
(t)
==
1
uniformly in ( 1 � � and in every finite t-interval. It then follows from the corollary to theorem 5 . 6. 1 that f ( t ) is infinitely divisible. n-+oo
Corollary to theorem 5 . 1 1 . 1 . Let f(t) be a self-decomposable characteristic function; then fc (t) is infinitely divisible. Let m < n; we re,vrite ( 5 . 1 1 .2) in the form k m k t f t f f( t ) . ) / ( IT ) ( ( k k ( k k IT 1 1 / ) ) n m k=m+ l n k= l m c Suppose that m increases as n increases in such a manner that n ·
=
->-
Some authors [e.g. B. V. Gnedenko and A . N. l(olmogorov (19 54) an d others] refer to this family as the "L-class". We do not use this te rminology in order to avoid c9nfusion with the 2' -class introduced in Chapter 9. ·
163
INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS
(ct)
as n -+ oo . The first factor then tends to f while the second factor tends to a characteristic function fc which by the argument used in the proof of the theorem is infinitely divisible. Our next aim is the determination of the canonical representation of self-decomposable characteristic functions f ( Since f is infinitely divisible, we can write it in the Levy canonical form (theorem and see that
(t),
(t)
t).
5.5 .2)
ita-a2t2/2+ I = : (eitu _ 1 - 1 �uu 2) dM(u) oo itu it u _ + I +o (e 1 - 1 + u 2) dN(u), where a2, M(u) and N(u) satisfy the conditions of theorem 5.5. 2 . Substituting tc for t in (5.11.3), we obtain after simple change of the variable of integration (5.11.4) log f(ct) ita1-a2c 2 t2/2+ I = : (eitu _ 1 - 1 �uu2) dM (:) + I +o (eitu _ 1 itu+u 2) dN (uc) where 0 < c 1o and oo (1 -c2\u3 (1 -c2)u3 a1 ca+c I - oo (1 + c2 u2)( 1 + u2) dM(u)+c J +o (1 + c2 u2)(1 + u2) dN(u). We see from (5.11.3) and (5. 1 1. 4) that (5 . 1 1.5) log ;g;) = ita2 -a2(1 -c2)t2j2 u _ u (e + I= : it 1 - 1 � u 2) d [M(u) - M (:)J + J :0 (eiru _ 1 - 1 �uu 2) a[N(u)-N (:)J where a 2 a-a1 • According to the corollary of theorem 5.11.1, (t) J(t)/f(ct) infinitely divisible ; therefore ( 5 .11.5) is its canonical representation and conclude from the uniqueness of this representation that M(u)-M( ujc) and N(u)-N(ujc) must be non-decreasing. Therefore (u1) -M (u1/c) M (u2) -M (u2/c) (5.11.6) {MN(v1)-N(v1/c) � N(v 2 )-N(v 2 jc) (5.11.3)
log _f (t)
==
a
==
00
0 and choose and so that = ea - b . We put = ea+h , = ea so that = b . It then follows from (5 . 1 1 .8) that = eb +h , N(eb+h) - N(eb) N(ea+h) - N(ea) . We write then =
f(t) /f(ct)
f(t) b vb v (u) u u2 , 2 N(u2/c)-N(u1/c) � N (u2) -N (u1). v2 h / a b c a v1jc e v2 c v1 N (e11) A(v); � A(b+h)-A(b) � A(a+h)-A(a) a+h x (say) or if we put b A(x) � �[A(x+h)+A(x-h)]. The function A(x) is therefore concave and has everywhere finite left hand and right-hand derivatives. The right-hand derivative never exceeds the left-hand derivative, and both are non-increasing as x increases. Since A(v) N(ev) we have A '(v) ev N '(ev). Putting u ev, we see that uN '( u) is a non-increasing function. In exactly the same way one shows that uM' ( u) is non-increasing. Suppose converseiy that the functions M(u) and N(u) have the property that uM '(u) and uN '( u) are non-increasing and that 0 < c < 1 . Then : M' (:) � uM'(u) for u 0 : N ' (:) uN '(u) for u > 0. From these inequalities we obtain (5 . 1 1 .6 ) by integration, so that the infinitely divisib le distributions determined by the functions M(u) and N (u) are self-de composable. We have therefore obtained the following result : Theorem 5 . 1 1 .2. An infinitely divisible characteristic function is self decomposable if, and only if, the functions (u) and N (u) in its Levy canonical representation have left- and right-hand derivatives everywhere and if the function uM' ( u) is non-increasing for u < 0 while uN '( u) is non increasing for u > 0. Here M'(u) and N'(u) denote either the right or left derivatives, possibly different ones at different points. Corollary to theorem 5 . 1 1 .2. All stable characteristic functions are self decomposable. =
=
=
=
=
0. 'fhe assumptions of the lemma imply that for n sufficiently large fr� # 0 for 0 � � Using ( iv) we see that
(t)
(m )
t a. J)l - I J.. (t) l 2] dt � z J : [l - J fn(t) j] dt 2Na (fn) · Moreover it follows from lemma 6.1.1 that J :a [1 - I j,, ( t) 12] dl = 2 J : [ 1 - I .fn (2t) I 2J dt � 8 J )1 - I fn ( t) / 2] dl. �
168 We combine the last two inequalities and conclude from assumption (6.1.2) of the lemma that a 2 r 2 J I lim / t) 1 dt 0. ( [ ] ,. n-+ oo J It is then easily seen that for sufficiently large fn ( t) 0 for 0 � t � 2a, so the argument which we used can be repeated. In this way we see that, for every T 0, (6.1.3) n-+limoo Jr T [ 1 - [fn (t) [ 2] dt 0. We denote by Pn(x) 1 -Fn( - x - 0) the conjugate distribution of Fn (x) and write (6.1.4) Fn(x) Pn(x) * Fn(x) for the symmetric distribution whose characteristic function is I fn ( t) [ 2 • We denote by 1 ) (x v2n J[ "'- oo e- u'/2 dy CHARACTERISTIC FUNCTIONS
=
0
#
n
>
=
0
=
=
=
the standardized normal distribution and consider the distribution defined by
(6.1.5) Gn (x) Fn (x) * (x) whose characteristic function is gn (t) e- t2j2 1 fn (t) 1 2 . From the inversion formula we see that sin tx - ' 2 f,. "' [ ! Gn (x) - Gn { x) n J - oo t e t / / (t) / 2 dt. Since Gn (x) is a symmetric distribution this can be written as Gn (x) - � ! J � si� tx e - t•/2 / f,. ( t) / 2 dt ! [ "' sin tx e - t'l2 dt 1 [ "' sin tx e - t'/ 2 [ f,. tW 1] dt. ( + / nJo t nJo t We write [ "' sin tx - t' /2 ,. 2 ! I,. (x) n J o t e [ / J (t) 1 - 1 ] dt =
=
=
=
=
=
and see that
T 0 tx e - t'/2 [ I /. (t) 1 2 - 1] dt t
Since for any ! r T sin
n Jo
>
�
L:J r T [1 - l In (t) ! 2] dt, n Jo
·
169
FACTORIZATION PROBLEMS-GENERAL THEOREMS
we conclude from
(6.1.3) that
lim
n-+ oo
so that
(6.1.6)
Lim n-+oo
In (x)
=
0
Gn (x) (x). =
It follows from the continuity theorem that lim gn = e- t2/2 lim J
(t)
n-+ oo
so that
fn (t) j 2 lim /fn (t) / 2 1 n-+ oo
e - t212
=
n-+ oo
and therefore Lim
=
(6.1.7) n-+ oo Fn (x) s(x). We write (6.1. 4 ) in the form P., (x) f " ro F.. (x-y)dPn ( Y) =
=
and get
P., (x) f� F.. (x-y) dPn( Y) � Fn (x-s) F"n (s) Fn (x - s)[1 -Fn( -�s - 0)] , where is an arbitrary positive number. Since by assumption x median of Fn (x) we see that Fn (x) � �Fn (x-s). We conclude from the last relation and (6.1. 7 ) that for any x < 0 0. F (x) (6.1.8) nlim n -+ On the other hand, we see by a similar reasoning that 1 - P.. (x) f [ 1 -Fn (x - y)] dPn ( Y) � � [ 1 - Fn (x+s)] that for any x 0 lim Fn (x) 1. ( .1. 9) I�'ormulae (6.1.9) and (6.1.8) imply the assertion of the lemma. ;;::
oo
=
e
6
;;::
>
n-+ oo
0 is a
=
oo
so
=
008
=
General decomposition theorems In this section we discuss three general theorems concerning the facto ri zation of distribution functions and characteristic functions. The first two of these theorems are due to A. Ya. Khinchine, the last is due to I-1 . Cramer. 6.2
1 70
CHARACTERI STIC FUNCTIONS
Theorem 6.2. 1 . Every characteristic function can be represented as the product of at most two characteristic functions which have the following property: one does not have any indecomposable factors while the other is the convergent product of a finite or denumerable sequence of indecomposable factors. Le t f(t) be an arbitrary characteristic function and denote the corre sponding distribution function by F (x). Since f(t) is continuous and f(O ) 1 , there exists a real a such that f(t) 0 if I t I � a ; in the follow (X. ing we fix such a value a and write Na (f) If f(t) does not have any indecomposable factors then the theorem holds. We suppose therefore that f(t) has indecomposable factors. Then (X/2 ; it is possible that f(t) has a prime factor p 1 (t) such that Na (p 1 ) it follows then from (iii) [see p. 1 67] that one can write f(t) P 1 (t) f1 (t) (X/2 while Na (f1 ) < (X/2. In this case we repeat the where Na (p 1 ) procedure with f1 (t) but use (X/4 instead of (X/2 as the lower bound for the Na-value of its prime factor. If f1 (t) has an indecomposable factor whose Na-value exceeds (X/4, then one obtains a decomposition f( t) P1 (t) P 2 (t) f2 (t) (X/4 (j 1 , 2) ( t ) while every indecomposable factor g(t) where Na ( P 3) of f2 (t) has the property that Na (g) < rt/4. In the case where no indecom posable factor with Na-value greater than rt/2 exists, we search for prime (X/4. In the case where such factors p(t) which satisfy the relation Na (p) factors exist one obtains a decomposition into at most four factors .f(t) P1 (t) Pn2 (t) f2 (t) \vhere 1 � n 2 � 3 . Here the p3(t) are indecomposable factors and satisfy the inequality Na ( P 3) (X/4, while every prime factor g( t ) of f2 (t) has the property that Na ( g ) < (X/4. We repeat this procedure and see that f(t) can be decomposed in the #
=
=
>
=
>
=
=
>
>
• • •
=
>
following manner :
(6.2. 1 ) are indecomposable factors such that Na (p3) > (Xj2k where the 3 (j = 1 , 2, of 1 � � 2 1 ) and where every prime factor has the property that Na (g) < rx j2 . It can happen that for some k � 1 the characteristic function/Jr (t) has no indecomposable factors. Then our process terminates and we see that the theorem holds. We must therefore prove the theorem only in the case
fk (t)
p (t) . . . , n1c ;
-
k
g( t)
terms "prime factor" an d ind ecomp osable factor syn onymously. (1") Note that accord i n g to our cons truction N(t (p1) > a./2 .
( tlfc ) We use
the
nk k
1 71
FACTORIZATION PROBLEMS-GENERAL THEOREMS
where the factorization process does not terminate ; the factors p 3 ( t) then form an infinite sequence. Since
� Na {p;) < Na =1
we see that the series
is convergent , so that the sum
(j )
k +m � Na {pj ) j= k +l
converges to zero as k tends to infinity ; this convergence is uniform in m ( m > 0). We now apply lemma 6. 1 .2 and see that there exist real numbers A v .v ' , such that lim eitAv.v' IT pk (t) = 1 v-+oo k= v uniformly in every finite t-interval I t I � and v ' > v. We write pk ( t) = Pk ( t) exp [iwk ( t)] and see that
T
tAv.v' + 2: w k (t) = 2nBv .v' (t) + o( 1 ) as V -+ oo k=v where Bv.v' ( t) assumes only integer values. The left-hand side of ( 6.2.2) is continuous, moreover Bv.v' (0) = 0 ; hence Bv.v' (t) = 0 for sufficiently large v and we have v'
(6.2.2)
tAv.v' + � wk (t) = o( 1 ) as v -+ 00 . k =v It is no restriction to assume that w7c ( 1 ) == 0 so that A v.v' ,,
� wk (t)
We see therefore that v'
k -= v
==
o( 1 )
=
o(
1) and
as v -+ oo.
li1n IT pk (t) == 1 v-+oo lc= v u niformly in I t I � and v ' > 7). rrhe infinite product
( 6.2.3)
T
IT P j (t) j= l 00
is then convergent ; let v( t) be its limit. It follows from ( 6.2.3) that
v(t) (•)
'fh is
cu n
be
Hl� l� ll
if
one
=
k
lim IT P3 (t) lc->-
00
j=
-
1
n1ultip lics each Pk (t)
by
cxp
[ - £t w1c ( 1 )] .
172
CHARACTERISTIC FUNCTIONS
where the convergence is uniform in every finite t-interval. We see then from the second version of the continuity theorem (theorem 3 .6.2) that is a characteristic function. Let s > 0 be an arbitrary positive number ; according to (6.2.3) we have
v(t)
k+m I IT l j=k+l
pj(t)- 1
< s
(m
>
0, I t I �
T)
if k is sufficiently large. Since one can conclude easily that
l fk (t)-fk+m (t) l < (m 0 , l t l � T). This means that the sequence {fk ( t)} also converges to a characteristic function. Denote u(t) = lim fk (t). It follows from ( 6.2. 1) that f(t) = v(t) u(t). The function u( t) has no indecomposable factor ; this follows easily from the fact that each indecomposable factor of u(t) must be a prime factor of fk (t) for all k. But such a factor cannot exist since the Na-values of the prime factors ofjk (t) tend to zero as k goes to infinity. This completes the e
>
k--"" 00
proof of theorem 6.2. 1 . The second decomposition theorem supplements the preceding result by characterizing the distributions which have no indecomposable factor.
Theorem 6.2.2. A characteristic function which has no indecomposable factor is infinitely divisible. Let f(t) be a characteristic function which has no indecomposable factors and denote by D f(t) = f1 (t)f2 (t) · · · fn(t) (D) an arbitrary decomposition of f ( t) where all fi ( t) are characteristic func tions. Suppose that a is a positive number such that f(t) 0 if I t I � a; we write then v(D) = max Na (fi) l The converse of theorem 6.2. 2 is not true . 'fhis will be shown later.
1 73
FACTORIZATION PROBLEMS-GENERAL THEOREMS
v
It follows from the definition of that there exists a sequence of decom positions say ) f ( t) = fn. l ( t)fn.2 ( t) · · · fn.kn ( t) ( n = 1 , 2, for which converges to so that
(Dn)
{Dn }, v(Dn)
.
v v v(Dn) < v + 1 /n (n = 1 , 2, . . .) . Let fin> ( t ) be the factor of Dn for which v(Dn) = Na (fin>) and write fr> ( t) for the product of all other factors of Dn . Then v Na ( fin> ) < v + 1 /n (n = 1 , 2, . . . ) (6.2.4) •
•
�
�
f { t) = fin> { t)f�n> { t). Let Fin) (x), F�n> (x) and F (x) be the distribution functions corresponding to fin>, f�nl and f respectively ; it is no restriction to assume that x = 0 is a median of F�n> (x). According to Helly ' s first theorem the sequence Fin> (x) contains a convergent subsequence, it is only a simplification of our nota tion if we assume that the sequence Fin> (x) itself is a convergent sequence . We prove next that the limit of this sequence is necessarily a distribution function ; this is established if we can show that for every 1J > 0 and suffi ciently large values of a > 0 and n one has Fin> { - a)
{a) > 1 - 1') .
We carry the proof indirectly and assume that one of these inequalities, for instance the second, is not satisfied for a > 0 and arbitrarily large n. Then for any b > 0 1 - F (b) = �
rX> oo [1 - Fin> (b -y)] dF�n> (y) J:l [1 - Fin> (b -y)] dF�n> (y)
� [ 1 - Fin> (b + 1 )] [ 1 - F�n> ( - 1 ) ] � � 1') . This however contradicts the assumption that F (x) is a distribution function. The proof of the inequality Fin) ( - a) < 1J for sufficiently large a > 0 and n is carried in a similar manner and so we have shown that
Lim Fin) (x) = F1 (x) n-+oo is a distribution function. We write f1 ( t) for the corresponding character istic function. We consider next the sequence Fkn> (x) ; it also contains a convergent subsequence and we use again the notation F�> (x) for this convergent subsequence. Let F2 (x) == Lim F�n> (x) ; n->-oo
'fhis cnn ulwtlys be accomplished by ll translation which does not affect the Na
...
valucs .
1 74
CHARACTERISTIC FUNCTIONS
we show that F2 (x) is also a distribution function. We have for any a > 0 and b > 0
[J a"'+ b [1 - Fin> (a -y)] dF�n> (y)
1 - F (a) �
� [l - Fin> ( - b)] [1 - F�n> (a + b) ] .
F"'or sufficiently large b we have Limoo Fin> ( - b) = F1 ( - b) < � n� and therefore 1 - F�n> (a + b ) � 2[ 1 - F (a)] . This indicates that the left-hand member of this inequality tends to zero as a increases ; in a similar manner one can show that F�n> (x) tends to zero as x goes to - oo provided that n is sufficiently large. Thus F2 (x) is a distribution function and n�oo is its characteristic function. It follows from (6.2.4) that f ( t) = !t ( t )f2 ( t ) while
Na (ft)
= 'V.
v
We next show by an indirect proof that < � Na ( f) . Let us therefore suppose that v � �Na (f) ; it follows from (iii) {page 1 67) that Na {/2) � v. If we decompose /1 and /2 once more, we obtain a decomposition
(D*)
f = g1 g2 ga g4
v(D*) < v. But this contradicts the definition
which has the property that of v, so we can conclude that
< � Na ( f) . Therefore there exists a decomposition of f ( t ) such that < � a ( f) . Each factor of is a characteristic function without indecomposable factors and we can apply the result to a factor of and see that < lNa ( f) . < We iterate this procedure to obtain the statement of the lemma. We proceed to prove theorem 6.2.2. Let {sn} be a sequence of decreasing positive numbers such that lim fn 0 . n�oo It follows from lemma 6.2. 1 that there exists, for each n, a decomposition t) = fn.l { t )fn .2 { t) (Dn) fn .1rn { t) V
D v(D) N
D
D
v �v(D)
=
f(
•
•
•
1 75
FACTORIZATION PROBLEMS-GENERAL THEOREMS
such that for j
=
1 , 2 , . . . , kn while
lim kn = 00 . n�oo Using lemma 6. 1 .2 we conclude that it is possible to find constants ('J..n.i { j = 1 , . . . , kn ; n = 1 , 2 , . . . ) such that lim {fn.i { t) exp [it('J..n .i] } = 1 n�oo where the convergence is uniform for 1 � j � kn and in every finite interval I t I � We write fn.i (t) = Pn.i (t) exp [iwn.i (t)] (6.2.5) f (t) = p(t) exp [iw(t)] where wn.i (0) = w(O) = 0. Then
{
T.
n�oo (uniformly in I t I �
hence also
lim [wn,j (t) - tw n.i ( 1 )] n� oo
Writing
we see that (6.2.6)
T
n�oo and for 1 � j � kn )· We see then that lim [wn,j ( 1 ) + ('J.,n,j] = 0, n-+oo
gn,i (t)
=
=
0.
]} Pn.i (t) exp { [w n,j (t) - tw n .A l ) + t k--; .
w( 1 )
t
and conclude that (6.2.7) lim gn,3 (t) = 1 ; n�oo the convergence here is uniform in I t I � I�" rotn the relation Ten
T and for j
TI Jn,j (t) j=l and from (6.2.5) and (6.2.6) we can easily see that J( t)
=
=
1 , 2,
. . .
, kn.
176 It follows from (6.2.7) that the assumptions of the corollary to theorem 5 .6.1 are satisfied ; we finally conclude from this corollary that f(t) is CHARACTERISTIC FUNCTIONS
infinitely divisible. The first two theorems discussed in this section show that there are three possibilities for the product representation of an arbitrary characteristic function ( :
f t) (I) f(t) has no indecomposable factor (in this case it is neces sarily infinitely divisible) ; (II) f(t) is the product of a finite or denumerable sequence of indecomposable factors ; (III) f(t) is the product of two characteristic functions /1 (t) and /2 (t) where /1 (t) has no indecomposable factors, while /2 (t) is the finite or denumerable product of indecomposable factors.
The decomposition is in general not unique ; this is illustrated by the example discussed in Section and also by the multiple factor (p . ization of the characteristic function defined by The first of these examples refers to a purely discrete distribution, while the function is the characteristic function of an absolutely continuous distri bution. In the next section we will find some further examples of multiple factorizations of absolutely continuous distributions. The converse of theorem is not true. We have already given (page an example of an infinitely divisible characteristic Section function which is the product of an indecomposable characteristic function and an infinitely divisible characteristic function. The characteristic function
5.1 104) g( t),
( 5.5 .12).
(5.5.12) 124,
5 .5)
6.2.2
p- 1 t (p > 1) f(t) p-e i =
of the geometric distribution which we considered at the end of Section permits an even more remarkable factorization. It is easily seen that
5.4
p - 1 t Jl p2k pe2it•2k . p - ei 1 =
oo
+
+
We see therefore that the characteristic function of the geometric distri bution is infinitely divisible but admits nevertheless a representation as a product of an enumerable sequence of indecomposable characteristic functions. Our next theorem indicates that the existence of infinitely divisible distributions with indecomposable factors is not a rare occurrence.
Theorem 6.2.3. An infinitely divisible character£stic function g(t) whose Levy canonical representation is determined by the constants a 0 and = a =
177
FACTORIZATION PROBLEMS- GENERAL THEOREMS
by the functions O < u < c N(u) {�(u - c) ifotherwise and M (u) 0 always has an indecomposable factor. Here k > 0 and c > 0 are arbitrary real constants. The characteristic function g( t ) is then given by fc (e1w - 1 - itu+ u 2) du. log g( t) k 1 Let be a real number such that 0 < s < � and introduce the functions if ( � - s) c < u < ( � + s )c -s (6.2.8a) cx1 (u) 1 elsewhere in (0, c) 0 outside (0, c) 1 f ( � - s) c < u < ( � + ) c { +s cx (6.2. 8b) 2 (u) - 0 if I u-c/2 1 � ) (6.2. 8c) log g; (t) k s: (euu_ t - �: 2 cx ( u) du (j 1 , 2). 1 Clearly k[cx1 (u) + cx 2 (u) ] N '(u) so that log g( t) log g1 ( t) + lo g g2 ( t) . (6.2.9) The function g2 ( t ) is, according to the representation theorem 5.5 .2, the characteristic function of an infinitely divisible distribution. We show next that g1 ( t) is also a characteristic function provided that is sufficiently small. We define a sequence of functions flv(x) by means of the recurrence relations {31 (x) kcx1 (x) {J,. (x) s :,,,fln -l (x- t) {Jl ( t) dt s: fln -l (X-t) {Jl(t) dt. 0 if We conclude from these relations and from (6.2.8 a) that f3n (x) either x 0 or x � nc and note also that I f3n (x) I kn cn - 1 . We remark that f3n (x) is the n-fold convolution of {3 1 (x) with itself so that J :" e t {J.. (x) k" [ s : eitre (1;1 (x) dxr· series � ( 1 /n !) f3n (x) n= 1 =
=
=
s
0
==
i
_
sc
;
=
s
=
=
=
B
=
=
=
=
�
�
i
'fhe
dx
"
=
00
is absolutely and uniformly convergent ; therefore
(6.2. 1 0)
s: L�1 �/ (x)] dx e1""
..
=
exp
[k J: eit" cx1 (x) dx] - 1.
1 78 We write
CHARACTERISTIC FUNCTIONS
c A. = k J rx1 (x) dx
and
o
and obtain from (6.2. 8c) and (6 .2.10)
o
X
1 + x 2 rx1 (x) dx
oo 1 g1 (t [ + J eitx ( i �n . Pn(x) ) dx] exp (-A.-itn). )
Let
c 1J = k J
=
0
n- 1
'
then
gt (t) = J'"' oo d 1 (x). In order to sho\v that (t) is a characteristic function we must prove that G1 (x) is a distribution function. Clearly G1 ( ) = 0, while we see from (6.2.8c) that G 1{ + ) = g1{ 0) 1. We must still show that G1{x) i& non- decreasing ; we do this by proving that 00� 1 n (x) I n . fJ is no n-negative for all x, provided that s is chosen sufficiently small. We first remark that {31 (x) is non-negative except in the interval (-�2 -s)c < x < ( � + s)c; moreover it is easily seen that {J 2 (x) tends to k ( c - I c - x I) uniformly in the interval 0 < x < 2c as s tends to zero. One can also show that {3 2 ( x) and {J ( x) as well as 1 1 (x) (x) + + {J fJI 2! 2 3 ! fJa (x) are non-negative for all real x if s is sufficiently small. Let s be such s0 a value. For n � 4 we can rewrite the relations defining the (x) in the form Pn + 2 (x) J :" Pn (x-t) {J2 (t)dt (n 2, 3 , . . .) and see that fln+ 2 (x) � 0 for all x and all n � 2 if s0 • Therefore ito: G
g1
oo
- oo
=
n -:-
-,
3
=
f3n
=
=
e =
G1 (x) is the distribution function whose characteristic function is (t). We show next by means of an indirect proof that g1 (t) is not g1infinitely divisible. Let us then assume tentatively that g1 (t) is an i nfinitely divisible characteristic function. 'fhen g ( t) admits a Levy and
1
FACTORIZATION PROBLEMS-GENERAL THEOREMS
N1 (t).
179
canonical representation with some non-decreasing function The derivative exists almost everywhere and is non-negative. of It follows from the uniqueness of the canonical representation that the relation ( ) = = N� ( ) ( ) � ( ) is valid almost everywhere in the interval (0, but this contradicts 6.2.8b . Hence cannot be infinitely divisible. It follows from theorem 6.2.2 that g1 must have an indecomposable factor, and we see from that this is ' also true for so that theorem 6.2.3 is proved. In exactly the same way in which we proved theorem 6.2.3 it can be shown that an infinitely divisible characteristic function whose Levy canonical representation is determined by the constants a = a = 0 and by the functions = < < 0 and = 0 and for or > 0 always has an indecomposable factor. We = 0 for < can therefore reformulate our result :
N� (t) N1 (t)
N' x k g1 (t)
( ) (6.2. 9)
(t)
M(u)
N(u) u -c u
x + kcx2 x kcx2 x c);
g( t),
M(u) k(u + c)
-c u
Corollary 1 to theorem 6.2.3 . Let f (t) be an infinitely divisible characteristic function and suppose that the functions M(u) and N(u) which occur in its canonical representation satisfy the following condition: there exist two positive constants k and c such that at least one of the relations M'(u) > k almost everywhere in ( - c, 0) or N' (u) > k almost everywhere in (0, c) holds. Then f(t) has an indecomposable factor. We consider only the case where the condition is satisfied in the interval (0, c). Then f(t) has an infinitely divisible factor of the form required by theorem 6.2. 3 , so that the corollary is established. Let f(t) now be a characteristic function which satisfies the conditions of this corollary. The function f(t) can be written as an infinite product f(t) = II [f(t)] 2-s. Each factor [f(t) ] 2-s also satisfies the conditions of the corollary, so that we co
s=l
obtain the following result :
Corollary 2 to theorem 6.2.3 . Suppose that the infinitely divisible character istic function f(t) satisfies the condition of the corollary 1 , then it is divisible by the product of an infinite sequence of indecomposable characteristic functions. We finally remark that the conditions of corollary 1 are satisfied by the Gamma distribution and also by all non-normal stable distributions. Generalizations of theorem 6.2.3 were given by R. Shimizu ( 1964) [see also B. Ramachandran (1967)] . We mention here only one of his results which shall refer later. ju11ction g(t) 'tht!ortJ1n o.2.4. An infinite!:>' to
we
di�visibh) rhararll)ristir
2ohose Lecz'y
1 80
CHARACTERISTIC F UNCTIONS
canonical representation is determined by the constants a 0 and the functions k(b-c) for 0 < u b N (u) k( - c) for b < u < c and M(u) 0 0 for � c always has an indecomposable factor. Here k > 0 is an arbitrary constant, while the constant c satisfies the inequality 0 < 2b < c. =
=
a
=
�
u
=
u
The proof is similar to the proof of theorem 6.2.3 . We select first a point such that 2 < < and a number s > so that 2b < - s) < ( + s) < . We define if < < -s elsewhere in outside (b, + s if ( - s) < < ( + s) cx 2 ( otherwise. The functions cx.1 and cx 2 determine again-according to formula (6.2.8c)-two functions g1 ( t) and g 2 (t) . The function g 2 (t) is an infinitely divisible characteristic function, while it can be shown that g1 is, for sufficiently small s, a characteristic function but not infinitely divisible.
b d c
d
0
d(1
d1 c d(1 -e) u d(1 + e) (b, c) c) d1 u d1
1 0 { 1 u) 0 (u) (u) =
(t)
Indecomposable characteristic functions We have already given several examples of indecomposable character istic functions, and we showed at the end of the last section that a rather wide class of infinitely divisible characteristic functions has indecompos able factors. This, however, is almost the only general theorem concerning indecomposable characteristic functions which we know at present. There is no general method for finding the prime factors of a given characteristic function ; our knowledge consists mostly of interesting special examples. In the present section we will make a few general remarks about prime factors and also list a number of remarkable decompositions. We consider next an arbitrary distribution function F( ) (which is not assumed to be of a pure type) and suppose that it is the convolution of two distribution functions F1 (x) and F2 6.3
x
(x): F ( x) J 00 F1 (x-t)dF2 (t). 00 Suppose that � is a point of increase of F1 (x) and a point of increase of F2 (x); it is then easy to see that � + is a point of increase of F(x). Similarly, if � is discontinuity point of F1 (x) and a discontinuity point of F2 (x) =
a
rJ
rJ
rJ
181 then � + 'f} is a discontinuity point of F (x) .(t ) It can also be shown that every discontinuity point ' of F (x) can be written in the form ' = � + 'fJ where � is a discontinuity point of F1 (x) while is discontinuity point of F2 (x) . Let {�i } and {'fJj} be the ( finite or enumerable) sets of discontinuity points of F1 (x) and F2 (x) respectively and denote the discontinuity points of F(x) = F1 (x) * F2 (x) by {'k } · Since the elements of the set {Ck } can be written in the form {�i +'fJ j } we see that every difference � i-�k must occur among the differences of ,i _ 'k at least as many times as there are
FACTORI ZATION PROBLEMS-GENERAL THEOREMS
'fJ
a
different n-values. From these considerations we obtain easily the following results :
(1) Suppose that all the differences Cj- Ck between the discontinuity points of a purely discrete function F (x) are different. Then F(x) is an indecomposable distribution function.(§)
(2) Suppose that the purely discrete distribution function F (x) has at least n 2 discontinuity points and that it is not possible to find n pairs of discontinuity points which differ by the same number. Then is indecomposable. (3) A finite distribution which has two discontinuity points, one at each extremity, is always indecomposable. The last statement follows from the earlier remarks and from the relations lext = lext lext rext rext rext
F(x)
\
[F1 * F2] [F1 * F2]
=
[F1] + [F2] [F1] + [F2]
F1 F2 •
which hold for the convolution of two finite distributions and In Chapter 3 (corollary to theorem 3 .3 .3) we showed that the number N of discontinuity points of a purely discrete distribution function has lower and upper bounds which are determined by the numbers of dis continuities of its factors. It can be shown that this corollary is also true if the distribution is not purely discrete, the same limits (n �N� are valid also in the general case. We then obtain by induction the following result : ( A purely discrete distribution which has exactly n 1 discontinuity points has at most n indecomposable factors. This maximum can only be attained if the discontinuity points are the ( n consecutive terms of an arithmetic series.
+ m- 1
4)
+ 1)
nm)
+
(-!') These statements follow from the inequality Jt1 (g + '1J + h 2) - F (f + 7J - h1) > [F1 (g + h 2 - k 2) - F1 (g - h� + k1)] [F2 ( 'YJ + k 2) - F2 ('YJ - k1)].
l h� rc
(§)
hl
0, h2
0, k l > 0, k 2 > 0. distribution function F (.x) is indecomposable if its characteristic
'"�e say that the fu nction is indecomposable.
>
>
1 82
CHARACTERISTIC FUNCTIONS
We consider next a purely discrete distribution whose discontinuity points are the consecutive terms of a finite arithmetic series. It is then no restriction to assume that this series consists of the integers 2, . . , n. In studying this class of distributions it is more convenient to use the probability generating function than the characteristic function. Let ak be the saltus of the distribution at the point k ( k the prob ability generating function is then the polynomial
0, 1, .
=
0, 1, . . . , n) ; 1).
(6.3.1) P (y) ak yTc (ak > 0, ak The substitution y eit transfortns P (y) into the characteristic function f(t) P (eit) ; the corresponding distribution function is given by =
� k=O
� k=O n
n
=
=
=
F (x)
n
=
� a�c s(x - k) . lc=O
Each decomposition ofj ( t) corresponds to a factorization of the generating function P (y) into the product of polynomials with non-negative co efficients. If no such decomposition exists thenf(t) is indecomposable. The number of factors of F (x) reaches its possible maximum if, and only if, the generating function has real, negative zeros. We assume next that all coefficients of the polynomial are equal and write
n
(6.3.1) n 1 Pn (y) n=
1
� yk k=o �
=
1 - yn n{ 1 - y )
for the generating functions of this sub -class. Since
Pnm ( Y)
=
Pn ( Y) Pm (yn)
Pm ( Y) Pn (Ym)
=
we see that the distributions of this class admit multiple decompositions, provided that the index is not a prime number. Using the present in the notation we could rewrite the example at the end of Section form P6 (y) P2 (y) P3 { y 2) P3 (y) P2 (y 3) . If
n
=
n
=
p�t p�2 •
5.1
=
• •
p�s ( CXt
+ +
h! ! cx2 !
!
CX 2
·
·
·
+
CX s
=
h)
is the decomposition of n into prime factors, then one can obtain in this way CX t
. . . CXs
different decompositions of Pn (y) into prime factors. M. Krasner B . Ranulac have shown that these are the only decompositio�s of
Pn (y) .
(1937)
The problem of decomposition into prime factors is therefore com-
Since the decomposition properties of distribution functions linea r transformations.
are
invarian t un d e r
183
FACTORIZATI ON PROBLEMS-GENERAL T.HEOREMS
pletely solved for the fa1nily of distributions with equal and equally spaced jumps. The indecomposable distributions which we have so far studied all had a finite set of discontinuity points. We show now that an indecomposable distribution can have an enumerable set of discontinuity points and can also be absolutely continuous or purely singular. Let { P v } be the sequence of prime numbers and suppose that the distribution function has its discontinuity points at �v = log Pv (v = . . . . It is then clear that the differences between discontinuity points are all different, so that is necessarily indecomposable. The following lemma will be used in our construction of a characteristic function which belongs to an absolutely continuous, indecomposable distribution functio11.
F(x)
1, 2, )
F (x)
Lemma 6.3 .1. Let p(x) be a frequency function which has a normal com ponent, then p(O) > 0. If p(x) has a normal component then it can be written in the form 2 x y) p(x) = �2n J -oo oo exp [ - � ; : J dF(y) where F(y) is a distribution function. Hence 2 ) + ( oo tt p(O) = �2n J - exp [ - 2 � J dF(y) > 0 oo a
a
a
a
and the lemma is proved. Let now
(6.3.2)
!( =
t) ( 1 - t 2) e-
Since
t s/2
=
_
d 22 (e -t"/2) . dt
1 J oo exp [itx - -x 2 dx, 2J V2n - oo see that j (t) = Jzn J oo oo x 2 exp [itx - �] It follows then that f ( t) is the characteristic function of the density 1 p(x) = vz; x 2 e - x e -t2 1 2 =
we
dx.
s; 2 .
remark by J. 1-Iadamard, which is app ended to the paper by l{rasner ltnn u l uc, t h i� p roh l t•tn wns also s ol ve d ind then
/1 (t) /2 (t)
/1 (t).
0),
/2 (t)
/2 (t) exp [ - #(eit - 1 )] = ell v'( l - A - 2) exp [ -# eit + v=�1 2 � j e2 m w VA
�
'--
'�
would be a characteristic function ; if we expand the right-hand side of this cq uation according to powers of eit we then see that the coefficient of eit is negative, so that exp [ - ft(eit _ 1 )] cannot be a characteristic function. ' rhis shows that (t) has no Poissonian component. 'fhe characteristic function is an example of a characteristic fu nction of an absolutely continuous, unbounded distribution function wh ich is indccon1posablc.
f2 (t) /2
(6. 3 .2)
1 86
CHARACTERISTIC FUNCTIONS
We next derive a theorem which enables us to construct certain interest ing examples of indecomposable distribution functions.
Theorenz 6.3 . 1 . Let F(x) be a distribution function and n be an integer. JiVrite (6.3 .5) Pn = F(n + 1 -0) - F(n-0) and define the distribution function (x) by (6.3 .6) F (x) = � Pnc(x-n) and introduce for for which Pn > 0, distribution functions Fn (x) by 0 ij X < 0 1 if n)-F(n-0) F(x � x< 1 (6.3 .7) ] + [ O P ifx ;?; 1. 1 Suppose that (a) P 2k + l = 0 for all k ; (b) p0 > 0; (c) ( x) is indeco1nposab le; (d) the distribution functions F2 k (x) have no com1non, non-degenerate factor. Then the distribution function F(x) is indecomposable. We give an indirect proof and assume that F(x) admits a decomposition (6.3.8) F = G * H, where G and H are both non-degenerate distributions. We introduce the quantities qn = G(n + 1 -0)-G(n-0) and r'll = H(n + 1 -0)-H(n - 0) and define the distribution functions G(x) and Gn (x) [respectively fi(x) and Hn (x)] corresponding to G(x) [ respectively H(x)] by replacing, in formulae (6. 3 .6) and (6.3 .7), Pn and F by qn and G [respectively rn and H] . In view of (b), we can assume without loss of generality that (6.3 .9) q0 > 0 and r 0 > 0. P
00
n= - oo
n,
n
P
We next show that (6.3 . 1 0) P = a * fl. It follows from (a) and (6. 3 .8) that
P21c = F(2k + 2 - 0) - F(2k -O) = f f dG(x) dH( y). 27c
<m + ·v 0 0.
rxk (2k 1)! c:� 'r/k = k k.
Hence all moments exist but do not satisfy (ii) . This distribution has another interesting property. Although its moment generating function does not exist, it is completely determined by its moments. It is known [see Shohat-Tamarkin ( 1 943), theorem 1 . 1 1] that a of its distribution function is completely determined by the sequence
1 2 / 7c diverges. Since rxk = (2k + 1 ) ! rx]; k= l 00
moments, if the sum �
n ow an integer We see then that
e - rlc K ' K�
k, I c i u > -l < er C' 1 f -c elvxldF (x) � _1 - e- (T-I YI >
- k 1)
- c- d
_ _ _
-=--
and apply an argument, similar to the one used above, to show that the i ntegral
r
"'
e l voo l
dF (x) exists and is finite.
Combining this with our
200
oo x dF (x), and therefore also J e'IIX dF ( x), l e earlier result we see that f " oo oo . Let = t iy , then the exists and is finite for all y such that I y I R oo z dF ( x) is also convergent for any t and I Y I R e integral f (z) = J oo and represents a regular analytic function, so that the sufficiency of our CHARACTERISTIC FUNCTIONS
'll
i
l
0 [respectively lm (z) 0].
0.
0
J : eixz dF (x) J: ei'"" dF (x) + f � ei"'" dF (x) ; =
the first integral is an entire function while the second is regular in the upper half-plane. We assume thatf(z) is an analytic characteristic function. Therefore f(z) is regular in a horizontal strip containing the real axis in its interior and admits in this strip the integral representation f(z)
J: oo ein dF (x).
=
Since f(z) is regular in the upper half-plane we see that the region of validity of this representation contains the half-plane lm (z) > in its interior. Therefore l f(z) I
for y
=
=
f oo
eitm - y - 1 log f(iy ) or that f(iy) < e - u(h - e)
F(x2)-F (x1) 0
=
e - y ( x2+ e> .
0 F 2 ) - F ( x1) 0
� for arbitrary B > and suffi 'J ,hcrefore e - 118 > == cien tly large y . This, however, is only possible if (x when eve r .t' 1 < .x2 = h 2e ; this means that (x) is bounded to the left and
t h at
-·
lext
[Jl]
F
� h.
20 2
CHARACTERISTIC FUNCTIONS
This completes the proof concerning distributions which are bounded hence from the left. However, we see from that log � also holds and we have = or the inequality � = lext
(7.2.2) h a [F] (7.2.3) lext [.F] = - lim sup ! log f(iy) . y-->- oo
We can therefore state
f(iy) ay, h a
Y
Corollary to thereom 7. 2 .2. If F(x) is a distribution function which is bounded to the left and has an analytic characteristic function, then (7 .2.3 ) lext [.F] = - lim sup Y! log f(iy); if F (x ) is bounded to the right and has an analytic characteristic function, then (7.2. 4) rext [F ] = lim sup y! log f( - iy). Y--* 00
1/--* 00
The proof has been given only for distributions which are bounded to the left ; the proof of the statements concerning distributions which are bounded from the right is quite similar and is therefore omitted.
Remark. Let k(x) be an arbitrary convex function such that k(O) = 0; it is then easily seen that k(x)jx is a non-decreasing function for x > 0. Let function which is regular in the half-plane f(z) belma(z)characteristic = y ( ex > 0). According to theorem 7.1 .4, the function log f(iy ) is a convex function of y for y > 0, and we conclude that [log f(iy)]/y is a non-decreasing function of y . Therefore it is possible to replace in the formula for the left extremity the lim sup by lim, and to write instead of (7 .2.3) (7.2 .3a) lext [.F] = - lim Y! log f(iy). - ex;
- 00
y
F(x)
These limits will of course be infinite if the distribution function is unbounded either to the left or to the right. If the distribution is finite, then we can combine the results of and its corollary and obtain the following statement : , theorem
F(x) 7.2.2 F(x) be a non-degenerate andfinite distribution function . 2.3 . Letfunction Theorem 7.cteristic (t) of F (x) is then an entire function of exTheon charat j>e 1: > 0 and of.forder 1 which has irifinitely 1nan)J zeros. Conp ential y
203
ANALYTIC CHARACTERI STI C FUNCTIONS
versely, an entire characteristicfunction of exponential type 1: > 0 and of order 1 belongs always to a finite distribution. Moreover the two extremities of F (x) are given by ( 7. 2 .3 ) and (7. 2 . 4). Let = lext [F ], b = rext [F], c = max ( I a 1 , j b !). We see then from theorem 7.2.2 that I f(z) I :( ecl zl , so that M(r; f) :( ecr. It follows from theorem 7.1.3 that f ( ) is an entire function of order 1 of exponential type not exceeding c. According to Hadamard's factorization theorem f( ) = G ( ) eaz where G ( ) is the canonical product formed with the zeros of f ( ) . Si�ce I f(t) I � 1 for real t, we see that G ( ) cannot be a polynomial and must therefore have infinitely many zeros. The second statement of theorem 7 . 2 .3 as well as the formulae for the extremities follow immediately from theorem 7 .2. 2 . Remark 1. A one-sided distribution does not necessarily have an analytic characteristic function. As examples we mention the stable distributions with exponent 0 1 and f3 = 1 which were treated in theorem 5.8.3. A specific example was given by formula (5.8.9). A one-sided distribution may have an entire characteristic Remark 2. function, and the order of this function can exceed 1. an example we a
z
z
z
z
z
z
< rx
0 ; then the integral J� exp (izx - kx1+cx) dx I( z) 1 + 1 /oc: and type k oc( :oc:) Hl/oc . is an entire function of order We expand the factor eizx into a power series, and since the order of integration and summation may be exchanged we see that n r (iz) I (z) ,,_L.o n ! J o xn e -kxl +cx dx . We introduce the new variable y kx1 +cx in the integral and obtain I ( z) Cn zn where n ) n i 1 + ( c n ! r 1 + ex (1 + ex) k1/ . We use Stirling's formula and the expressions (D4) and (D S ) of Appendix and obtain the statement of the lemma. D The order and type of entire characteristic functions depends on the =
r
p =
00
=
tl
l
00
=
=
=
n
� 00
n =O
=
"tail behaviour" of the corresponding distribution function. In order to study this behaviour it is convenient to introduce three functions. Let be a distribution function ; we write for > 0
x
T(x) 1 -F(x) + F(-x) log [ T (x)]- 1 T1 ( ) xl +cx ( ex > 0 ) (7 .2.5) log log [T (x)] - 1 T2 (x) log x We note that T1 (x) depends on the positive parameter ex. X
=
=
=
F(x)
ANALYTI C CHARACTERISTIC FUNCTIONS
205 > 0.
Lemma 7.2.2. Let F(x) be a distribution function and ex 0 and k Suppose that there exists an x0 > 0 such that1 T(x) � exp ( kx +cx) for x � X 0• Then F(x) has- an1 entire characteristic function f(z) which is either of order equal to 1 + ex and type � ex[k1/cx (1 + ex)1 + 1/cx] - 1 or of order less than 1 + ex -I. Let A > x0 and r > 0 ; we see then (integrating by parts) that AI e'"" dF(x) = - I A e'"'d [ 1 -F(x)] Xo Xo A A o r erx [ 1 -F(x0)] - e [ 1 - F(A)] + r I Xo [ 1 - F(x)] erx dx. We let A tend to infinity and conclude from the assumption of the lemma that I 00 e'"' dF (x) [ 1 -F (x0)] + r I 00 exp (rx - kx1 +ex) dx. Since r· e'"'dF(x) e'"'• F(x0), we have I 00 e'"' dF(x) e""• + r I � exp (rx-kx1 +cx) dx. (7.2.6) 00 Similarly one can show that (7.2.7) I e- ·"'dF(x) e'""' + r Jr o exp (rx - kx1 +cx) dx. Since M (r, f) = m ax [f(ir),f(-ir)] we see from (7.2.6) and (7.2.7) that M(r, f) e'""' + r I� exp (rx -kx1 +cx) dx. >
-
1:
=
� e"""'
�
�
�
oo
�
oo
oo
�
- oo
�
The statement of lemma 7 .2.2 follows easily from lemma 7 .2. 1 and from the last inequality. .�
'
Lemma 7.2.3 . LetA F(x) be a distribution function with characteristic and let 0, > 0 be two constants. Suppose that there exists functionf(t) a constant R such that M(r ; f) exp [Ar1 +1l] for r � R. Then lim inf T2 (x) 1 + l ift and T(x) exp { -x1 +p-l- e) .for any 0 and sufficiently large x. from (7. 1 .6) that for > 0 and r � R 11 (x) 2 ( rx Ar1 +tt) . >
#
�
�
x� oo
�
e >
We sec
x
�
exp
-
+
206
CIIARACTERI STIC FUNCTIONS
x ;:::: x0 2R�J and {�x)1/11- so that r �1 R; then we get T(x) � 2 exp [ - (2 -A) ( � x) +11--1] and we conclude from formula (7 .2.5) that lim inf T2 (x) ;:::: 1 + 1 / For any > 0 and sufficiently large x, one then has 1 T2 (x) ;:::: 1 + - - s . Using again (7. 2 .5), one obtains the statement of the lemma. We study next entire characteristic functions of order greater than 1. Theorem 7.2.4. The distribution function F(x) has an entire characteristic function f(z) of order 1 + cx -1 {ex > 0) and of intermediate type 7: if, and only if, the following two conditions are satisfied: ( cx7: - ) ( ) hm mf T1 (x) (1 + IXy +oc (ii) T(x) > 0 for all x > 0. We first prove that the condition is necessary, and assume thatf(z) is an entire function of order 1 + cx- 1 and finite type > 0. Clearly (ii) is necessary, since T(x) 0 means that F(x) is finite, so that f(z) would have order 1. Moreover it is possible to find for any > 0 a value R R( ) such that (7.2.8) provided that r � R. It follows from ( 7.1.6 ) and (7. 2 .8) that (7.2 .9) T(x) � 2 exp [- rx + ( 7: + s) r1 + - ] for x > 0, r � R. (�Y;"' and Let a be an arbitrary positive number and let x axcx ; then r ;:::: R. It follows from (7 .2. 9) that 1 ] log - log 2+x� + 1 [a - ( + s ) T(x) We put
r =
=
,u .
X-HO
e
,u
1 •
•
•
=
x� oo
I
cx
=
e
cx
1:
=
1
�
r =
�
so that
x (7.2.10 ) as
-+ oo .
Therefore lim inf x- ->-
oo
T1 (x) � a-(7: + s)a1 +cc1 •
1:
x0 =
a1 + cx-l
e
207
ANALYTI C CHARACTERI STIC FUNCTIONS
This relation holds for any a > 0 and in particular for that value of which maximizes the right-hand side of (7.2. 0 , that is for
a
1) a = {ex/[{ r +s)(1 + ex)] }ct. We substitute this value of a into (7.2. 1 0 ) and get 1 ex( r s) ct ] + [ . lim inf T1 (x) l +ct (1 +ex) Since s > 0 is arbitrary we see that 1 ( exr)ct li m inf T1 (x) (7. 2 . 1 1 ) (1 + r:x) We show next by means of an indirect proof that the inequality sign cannot hold in (7 .2. 1 1 ) . Suppose therefore that -r1 ( ex )ct y lim inf T1 (x) > ( 1 + ex 1 ct ( exr ) Then it is possible to find a k > (1 + y + such that T1 (x) ;;:,: k for x sufficiently large. Using (7 .2.5), we see that T(x) � exp ( - kx1 +ct) for suffi ciently large x and conclude from lemma 7.2.2 that (7.2. 1 2) M ( r ; f) � exp [( r ' + s ) r1 +ct-l ] [k1 11ct(1 + r:x)1 + 1/ct] - 1 ex < r. for any s > 0 and sufficiently large r, where r ' Since the order of f(z) is, by assumption, 1 + ex - , it follows from (7.2. 1 2) that the type off (z) is at most equal to r', hence less than r. This contra dicts the assumptions of the theorem ; therefore the inequality sign cannot hold in (7 .2. 1 1 ) , so that the necessity of ( i) is established. We still have to show that conditions (i) and (ii) are sufficient. Clearly (ii) implies that f(z) is not a function of exponential type and that T1 (x) is defined for x > 0. Let k < (exr - 1)ct { 1 + r:x) - 1- ct. (7. 2. 1 3) x1 (k) such that T1 (x) k for In view of (i) , there exists a value x1 x1 . It follows from (7.2.5) that T(x) � exp ( - kx1 +ct) (7 .2. 1 4) X u and we conclude from lemma 7.2.2 that f(z) is an entire func for tion whose order and type r' are such that either 1 + ex - 1 and r' � exj [k11cx (1 + r:x)1 +11ct] (7.2. 1 5) x� oo
>
x� oo
�
l +oc "
+ oc "
x� oo
cz
oc
=
x
=
�
x
or
�
�
p
p =
(7.2. 1 6) We sh o w next that (7.2. 1 6) cannot hold . We give an indirect proof and 'assu n1c therefore tt�n tativcly the val id ity of (7.2. 1 6). It is then possible to
208
CHARACTERISTIC FUNCTIONS
oc such that M(r ; f) exp (r1 + 11Y) for sufficiently large r, say r R. It follows from this inequality and lemma 7 .2. 3 that for any > 0 T (x) � exp ( - x1 + y-e),
find a number y >
�
s
�
provided that x is taken sufficiently large. We again apply lemma 7.2.2 and see thatf(z) is an entire function whose order p cannot exceed 1 + (y - s) - 1 • Since s is arbitrary, we see that p � 1 + y - 1 < 1 + ex - • But then M(r ; f) � exp and we see from lemma 7 .2.3 that lim inf T2 {x) � 1 + y > 1 + ex . (7.2. 17)
1 (r1 +"-1) ,
4
We also see from (7. 2 . 1 ) and (7. 2 . 5 ) that T1 (x) (7 .2. 1 8) x Ts<x> - < 1+ . Since ex and r are finite and positive , (i) implies that lim inf =
� 00
T1 (x) is finite
and positive. Equation (7 .2. 1 8 ) indicates that this is only possible if (7.2. 1 9) lim inf T2 (x) = 1 + ex.
6
Relation (7 .2. 1 7), derived under the tentative assumption (7. 2 . 1 ), con tradicts (7.2. 1 9), so that necessarily (7 .2. 1 5 ) is valid and p = 1 + ex - t, as stated in the theorem. Since is only subject to condition (7.2. 13) but is otherwise arbitrary , we deduce easily from (7.2. 1 5) that (7 .2.20) 7:1 � 7:. We show, again by an indirect proof, that the inequality sign in (7 .2.20) leads to a contradiction. Suppose therefore that r ' < r. Then there exists a r ' ' such that r' < r ' ' < r and M ( f) � exp ( 7:" for sufficiently large The last inequality has the same form as (7.2.8). We use the reasoning which led from (7.2.8) to (7.2. 1 1 ) and see that
k
r.
r1 +cx-1)
r;
( ex/ r " ) \1. ( ex/r) . . h��f Tl (x) � > {1 + {1 + This contradicts assumption ( i) of the theorem, so that r ' = 1: and the proof is completed. It is also possible to derive conditions which assure that a distribution function has an entire characteristic function of a given order greater than 1 and of intermediate but unspecified type, or of maximal or minimal type.
ocy +cx
oc)H"' . r�-
209 Theorem 7.2.5. The distribution f nction F(x) has an entire characteristic u function f(z) of order 1 + cx- 1 (ex > 0) if, and only if, the following two con ditions are satisfied: (i) lim inf T2 (x) = 1 + oc (ii) T(x) > 0 for all x > 0 . In view of theorem 7.2.4 it is clear that the conditions are necessary. To prove that they are sufficient, we note that (i) implies that T(x) � exp ( x1 + e< - e) fot' any > 0 and sufficiently large x. Using the argument which we employed in the proof of theorem 7.2.4, we can show that f(z) has order 1 + cx. - 1 • Theorem 7.2.6. The distribution function F(x) has an entire characteristic function of order 1 + cx- 1 (ex > 0) and of minimal type if, and only if, the following three conditions are satisfied: (i) lim inf T2 (x) = 1 + ex (ii) T(x) > 0 for all x > 0 (iii) lim T1 (x) exists and lim T1 (x) = + Theorem 7.2.7. The distribution function F(x) has an entire characteristic function of order 1 + cx- 1 (ex > 0) and of maximal type if, and only if, the following three conditions are satisfied: (i) lim inf T2 (x) = 1 + ex (ii) T(x) > 0 for all x 0 (iii) lim inf T1 (x) = 0. ANALYTI C CHARACTERISTIC FUNCTIONS
s
-
oo .
x---+ oo
>
It is also possible to obtain results concerning distributions whose characteristic functions are entire functions of order 1 . The method of proof is similar to that used in proving theorem 7.2.4. We therefore list here only the relevant results.
Theorem 7.2.8. 1The distribution function F(x) has an entire characteristic function of order and maximal type if, and only if, (i) T(x) > 0 for all x > 0 (ii) lim T2 (x) exists and lim T2 (x) = Theorem 7 .2. 9 . An entire function of order 1 and minimal type cannot be a characteristic function. +
oo.
,_rhe ]ast theorem is only a reformulation of a result from the th eory of en tire functions which asserts that a non-constant entire function of at
210
CHARACTERISTIC FUNCTIONS
most first order and minimal type cannot be bounded on some line [ see B. Ya. Levin ( 1964) , p. 5 1] . For a detailed proof of theorems 7 .2.5 to 7 .2.9 we refer to B . Ramachandran ( 1962). Order and type of entire functions provide means of studying their growth. This study can be refined by introducing proximate orders and types with respect to proximate orders [ see e.g. Levin ( 1964), pp. 3 1 :ff. ] . It is also po3sible to investigate the behaviour of characteristic functions having given proximate orders. For these studies we refer the reader to H. J. Rossberg (1966), (1 967a), (1 967b ). We finally remark that there exist entire characteristic functions of infinite order. Let f ( t ) be an arbitrary characteristic function ; it follows from lemma 5 .4. 1 that exp [f( t) - 1] is also a characteristic function. We define the sequence of functions f( t) /(1) ( t) (7.2 .21) f ( t) = exp [f(n - t) ( t) - 1] = 2, 3 , . . . ) and see, again using lemma 5 .4. 1 , that all functions of the sequence f< n> ( t) are characteristic functions. Suppose now that f ( t) is an entire characteris tic function of finite order, then the functions f ( t) are entire functions of infinite order if > 1 . As an example we mention the sequence of functions which starts with /(1) ( t) = eit ; this yields a sequence of entire characteristic functions of infinite order of more and more rapid growth. The second function in this sequence is the characteristic function of the Poisson distribution.
{
=
(n
n
7.3
Criteria for analytic characteristic functions In Chapter we discussed various criteria for characteristic functions. We have seen that the necessary and sufficient conditions developed by Bochner, Cramer and Khinchine are not easily applicable. It is therefore desirable to derive less general results which are applied more readily. These results are usually restricted to certain classes of functions ; in this connection the problem arises whether it is possible to characterize those functions which are regular in a (complex) neighbourhood of the origin and are characteristic functions. This problem is still unsolved, but a number of results , giving sometimes necessary and so1netimes sufficient conditions for analytic functions to be characteristic functions, were found. The present section deals with these criteria. In some instances we only mention conditions and give appropriate references, but a very important criterion for a class of entire characteristic functions will be studied in detail. We note first that some of the results treated in Section 7 . 1 can be re garded as criteria for analytic characteristic functions. 'l.,hus theorems 7 7 . 1 .2 and thei r corollaries give necessary conditions which a function
.1.1 ,
4
2 11
ANALYTIC CHARACTERISTI C FUNCTIONS
regular in a neighbourhood of the origin must satisfy in order to be a characteristic function. The same is true of theorem 7. 1 .4 or of P. Levy's result (theorem 7. 1 .3) that a non-constant entire function of finite order must have at least order and must have infinitely many zeros if its order is equal to It is easy to establish a condition similar to the one listed in theorem Let be an analytic characteristic function ; then
1
1. 4.1.2. f (z) Re [f(iy) - f(t + iy)] = I : "' e - ( 1 - cos tx ) dF(x) "' 2 e - ( 1 - cos 2tx)dF(x). � � J "'"' e - sin txdF(x) = 1 I "' '""
�� 0 such that Am (�m) > 0 while Bm (�m) 0. Lemma 7.3 .2. Let m = 3 and 0; then there exists a �� > 0 such that = 0 then there exists a A 3 (�;) < 0 and B 3 (��) 0. If m = 3 and �� > 0 such that A3 (��) > 0 and B3 (��) 0. �
=I=
=I=
?'s
y3
=1=
=1=
y3
In order to prove these statements we study the polynomials V8 (�) and W8 (�) and show that they can be expressed in terms of Chebyshev poly notnials or trigonometric functions of an auxiliary variable. ('t) f1'1 is here II
nnd in the
following the positive square root of e.
216 We consider the expression (1 +i v� ) s, where s is a positive integer and I � I < n /2. Then � 0, and se � = arc tan v� with (7.3.13 ) (1 + i v�) s = (1 + �)s/2 (cos s� + i sin s�). For s 2 we expand (1 + i v� ) s according to the binomial theorem and obtain (7.3 .14) ( 1 + i v�)s = 1 -�Vs (�) + i WWs (�) . We note that (1 +� ) 812 = (cos �) - s and get from (7.3.13) and (7. 3 .14) 1 - �V. (�) = (cos sf)!(cos cfo)• = (1 + �)•12 T. ( v'(l\ �)) (7.3.15) WW. (�) = (sin scfo)/(cos cfo)" = (1 +�)·12 Us-1 ( v'( l\ �)) where T8 (x) = cos (s arc cos x) and Us - 1 (X) = sin (s arc cos2 x) v'(1 -x ) CHARACTERISTIC FUNCTIONS
�
t
�
are the Chebyshev polynomials of the first and of the second kind re spectively. We introduce for the sake of brevity the notation y<m - 1 )/2] = { {) )[m/2 ] = and express the functions and If in terms of the variable we write = tan 2 � tan 2 = = Bm (tan 2 then we get from and sin cos r =
(7.3.16) {Y ( -- 11 �m Ym �Am (�) Bm (�) � (7.3.17) C(�) Am ( �) D(�) �), (7.3.11), (7.3 .15) (7.3.16) m� +� (7. 3 .1 8 ) C(�) [ 1 _ {COS m� �)mJ {COS �)m (7.3.19) D(�) = y {CsinOS mcfo�)m + {CcosOS mcfo . m �) We next prove lemma 7.3.1 by showing that it is always possible to find a �0 such that C(�0) > 0 while D(�0) 0. We give the following rules for the selection of � 0 : (I) If y > 0 and � 0, select �0 so that nj2m < �0 ' < njm while tan m�0 �jy . (II) If y > 0 and � < 0, select �0 so that njm < �0 < S nj4n-t while tan m�0 - fljy. {)
=1=
� =I= =I=
217 (III) If y = 0 and � > 0, select �0 so that nj2m �0 njm. (IV) If y = 0 and � 0 , select �0 so that njm �0 5nj4m. (V) If y 0, select a value �0 which satisfies the following three conditions : 2n n (a) m �o m (b) tan m�0 -�/m y, {c) h(�0) = y( cos �0- cos m�0) +� sin m�0 > 0. We must still show that it is possible to select �0 in case (V) so that condition (c) is satisfied. We first observe that h( �) = y( cosm � - cos m�) in m� is a continuous function and that h(2njm) > 0. Hence the +� s function h (�) is positive in some neighbourhood of � = 2njm, so that a selection in accordance with (c) is possible. The assumption em 0 implies that y and � cannot vanish simul taneously, so that our selection rule covers all possibilities. Using this fact as well as the assumption m ;:::: 4, it is easily seen that the value �0 whose selection we have just described satisfies the conditions C(�0) > 0 and D(� o) 0. But then it follows from (7.3.17) that �m = tan 2 � 0 satisfies the assertion of lemma 7.3 . 1 . We next prove lemma 7.3 .2. We see from ( 7 .3 .11) that A 3 (�') = - 3y3 �' - �3 V(�')(3 - �' ) B3 (�') = - r3 v(�')(3 -�')- �3 (1 - 3 �'). If y 3 0 and y 3 �3 0 we choose �� > 3, and if y 3 !5 3 ;:::: 0 we choose 0 �� 3. If y3 = 0 and< �3 > 0 we select �� > 3 ; if �3 0 we select 0 �� 3. Obviously it is possible to select �� in agreement with this rule so that B 3 ( ��) 0. This completes the proof of lemma 7.3.2. In the following we assume that m ;:::: 3 and choose �m and �� in accord ance with lemma 7.3.1 and 7.3.2 respectively. We write (7.3.20) Av = Av (�m), Bv = Bv (�m) and obtain from ( 7.3.12) (7.3.2 1) A 1 [yy(�m), y] = Am ym + Av yv. Now let m 4 ; then A I [Y v(�m), y] = Am ym [ 1 + o( 1)] as y We see from lemma 7.3.1 that Am > 0, so that A 1 [ y v( �m) , y] is positive sufficiently large positive values of y . We consider next the case m = 3 and write = sgn y = y!I y I · We choose so that ey3 0. Then A 1 [ y v'(�:; ) , y] = A3 s! y i 3 +- A2 y 2 + A1 s! y l = A 3 ei Y I 3 [ 1 + o(1)] Iyj ANALYTI C CHARACTERISTIC FUNCTIONS
-
0. Let () be an arbitrary real number, then there exists an integer g such that 0 -J- 2ng � B( y 0) < 0 + 2n(g + 1).
ANALYTIC CHARACTERISTIC FUNCTIONS
219
yy0 0 y y, B(y) y1 B(y1) y1 B(y1) y1 0, y1 ! A1 [y1 y1] 0, B( y1) - {31 [ y1 y1] .1. y* y 1 t* y1 y
We consider from now on only such values of for which > and l I � l Y o I · For such values of is either monotone increasing or monotone decreasing. In the first case we can find a real number such for which that = () + 2(g + 1 )n ; in the second case there exists a == 0 + 2gn . Since I we see from lemma � I Y o I and Y o > V�m' V�m ' () is an 7.3 .3A that > while () = integer multiple of 2n. To complete the proof of lemma 7. 3 . 3 we need only put an�. = v'�m· We are now ready to prove theorem 7.3 = Let
v= O be a polynomial of degree m > 2 ( 0) and K;; 1 = en (c0). We carry an indirect proof of the theorem and suppose therefore that In (t) == Kn en [P(t)] is a characteristic function. The function In ( t) agrees for real values of with the function A (z) == Kn en [P(z)] so that it is an entire characteristic function. From now on we consider this characteristic function also for complex values of the argument = t + iy and apply the ridge property (theorem em
=I=
z
z
7 . 1 .2) of analytic characteristic functions. This theorem indicates that necessarily
(7 • 3 • 23 )
t
R( Y ) =
t,
ln (t +iy) "' fn (iy)
� l
for all real and y. We now introduce the functions (7. 3 .24) == ( v == 2, where == and note that ) and that = exp We obtain easily from definition (7.3 .24) of the functions the recursion formula (7. 3 .25) = exp ( = 2, . . . , n). We now introduce the functions ( 7 .3 .2S a) = ) ( = 2, 3 , and write for the imaginary part of for the real part, cp1, (z) , so that ( 7 .3 .25b) = ) ( = ·
Kv [ev (c0)] - 1 (O) 1 1, . . . , n l v 11 (z) l (z)[4>1 (z)] . v {Kv-\ [lv- l (z) - 1 ]} v lv (z) ...,n . c/Jv(z) Kv!1 [ lv- 1(z)- 1 ] v rt.v (t, y) fJv (t, y) 4>v (z) rt.v (t, y) + if3v (t, y) v 1, 2, . . . , n (7 . 3 .2S c) lv (z) = exp [c/>v (z)] (v = 1 , 2, . . . , n) . (7 .3 .26) � (p1, iit.,,) ( == 2, . . . , n ) . Hnd
We set
K;- 1
cxp
-1-
·v
220
CHARACTERISTIC FUNCTIONS
v-\ = ev- l (c0) we see that K:;; 1 = exp (Kv \) or exp (Pv + i.Av ) = e 2 ( Pv- t + iAv- t ) , therefore (7.3 .27) Pv + iAv = exp (Pv- t + i.Av_ 1 ) + 2gv ni where gv is an integer. It follows from (7 .3 .27) that (7. 3 .28) Av = exp ( Pv -t) sin A v -1 + 2gv n. We combine (7.3 .2S a), (7 .3 .2Sc) and (7.3 .26) to get �v (z) = exp [pv -1 + iAv -1] {exp [cxv -1 (t, y) + iPv -1 ((t, y)=] 2,- 13}, Since K
v
)
. . . , n .
We separate real and imaginary parts in the last formula and obtain re cursion formulae for and cos (7.3 .29a) = exp [Pv -1 - exp (Pv - 1 ) cos (v = 2, 3 , . . . , n) (7.3 .29b) sin = exp [Pv -1 - exp (Pv - 1 ) sin (v = 2, 3 , . . . , n) . We now define the functions y) (v = 1 , 2, . . . , n ) = and see from (7 .3 .29a) that (7.3 . 30) = {exp COS [Av-1 + exp [Pv -1 + (0 , - cos (0 , v = 2, 3 , . . . , n . We apply lemma 7.3 .3 and select e = A 1 • Then it is possible to find a pair such that of real numbers (7. 3 .3 1 ) A1 >0 while (7.3 .32) = 2g1 n (g 1 integer). We show next that a similar relation holds for all functions fJv namely = 2gv n ( 7.3 .33) where are the numbers determined according to lemma 7.3 .3 and used in (7.3 .3 1) and (7.3 .32) ; is given by (7.3 .28 ) and gv is an integer. We prove (7.3 . 33) by induction. Formula (7.3 .32) indicates that (7. 3 . 3 3) is valid for v = 1 ; we suppose now that it holds for all subscripts inferior to v. We then have in particular = 2g n . Substituting this into (7 .3 .29b) and using (7 .3 .28) we see that sin = - exp = - Av + 2gv n . r-f hus (7. 3 .33) is generally valid.
rxv (t, y) Pv (t, y): + (Xv -1 (t, y)] A [Av -1 + Pv -1 (t, y)] (XV (t, y) v -l + rxv -t (t, y)] A [Av -1 +Pv -t (t, y)] Pv (t, y) v -l Av (t, y) rxv (t, y) - rxv (O , Av (t, y) [Av -1 (t, y)] y) Pv -1 (t, y)rx] - 1 y)] [Av -1 + Pv -l ]} ( v ) t* , y* (t*, y*) P1 (t*, y*) + A1 (t, y,) Pv (t*, y*) + Av t*, y* Av Pv -1 (t*, y*) + Av -1 v -l (Pv -t) Av -l Pv (t*, y*)
221
ANALYTIC CHARACTERISTIC FUNCTIONS
We see from (7. 3 . 30) and (7.3 .33) that A ( , * ) = {exp
[Av_1 (t*, y*)] - cos [Av -t + f3v -expt (O, y*)1]}+ {0, y*) [ v - !Xv -1 ] for v = 2, 3 , . . . , n. From this formula we see that the relation Av -1 (t*, y*) > 0 implies that Av (t*, y*) > 0. We can therefore conclude from (7.3 .3 1 ) that {7.3 .34) An (t *, y*) > 0 . We defined earlier the function R(t, y) as fn ( t + i ) . R(t' Y) = v t* y
X
P
y
fn ( iy) it follows from (7.3 .25b) and (7. 3 .25 c) that ( = exp {An ( We have therefore determined a pair of real numbers such that (7.3 . 35) ) > But this contradicts (7. 3 .23) which must be satisfied if fn ( ) is a charac teristic function. This contradiction completes the proof of theorem 7. 3 .3 since it shows that fn ( cannot be a characteristic function if m > 2. In the case where m � 2 the iterated exponentials fn ( ) = n can be characteristic functions. The function f1 ( ) = exp where {2 a1 and are both real, � 0, is a characteristic function (of a normal or of a degenerate distribution). It follows from the recursion formula (7.3 .25 ) and from lemma 5 .4. 1 that fv (z) as defined by (7.3 .24) is a charac teristic function for all values of v . We note that for m == 1 and 1 = we obtain for n == 2 a Poisson distribution and for == 3 a Neyman type distribution , fa ( ) = exp {,u exp We have already mentioned that Marcinkiewicz derived a particular case of theorem 7. 3 .3 in a different manner. He obtained it as a special case of a 1nore general theorem which gives a necessary condition which an entire function of finite order must satisfy if it is a characteristic function. We n ow state this theorem of Marcinkiewicz. '
R t, y) R(t*, y* 1. t)
a2
a2
n
t
t, y) }.
t
t*, y* t
t e [P(t)] [ - a2 + ia1 t]
c 1 A [ (A(eit _ 1 )) - 1 ] }.
An entire function finite order whose exponent of to1l7)ergence is less than cannot be a characteristic function. '/,heorem 7.3 .4. of p > 2 p1 p l n the proof of theorem 7.3 .4 we use a number of theorems from the theory of functions of a complex variable. The results needed may be fo und, for i nstance, in Copson (1 935), pp. 1 65-175 .
222
CHARACTERI STIC FUNCTIONS
Letf(z) be an entire function of finite order p. By Hadamard ' s factoriza tion theorem we can write f z) in the form (7.3 .36) f(z) = G (z) exp z) where (z) is the canonical product of the zeros of f(z) and where H (z) is a polynomial of degree m � p. We denote by p1 the exponent of con vergence of the zeros of f z) ; it is easily seen that p = max (p1, m) . If p1 < p then necessarily p = m. It is known that the order of a canonical product equals its exponent of convergence. Let (z) be a canonical product of order p1 ; then for any s > 0 the modulus I (z) I � exp z IPl + provided that I z I is sufficiently large. We will also use the following result which is due to E. Borel. If (z) is a canonical product of order p1 and if s is an arbitrary positive number, then there exists an infinite number of circles of arbitrarily large radius on each of which the inequality I ( ) I > exp ( - I z I Pl + e) holds. Let z = and denote by r = z I = We see then that there exist arbitrarily large values of r such that I I > exp ( - rP 1+ e). On the other hand we know that for arbitrary s > 0 and sufficiently large I � exp (rPl+e). We combine the last two inequalities and see that there exists an increasing sequence {r�c} of positive real nun1bers such that lim rk = oo
(
G
[H ( ]
(
G G
[I
�],
G
Gz
Vt 2 +y 2 •
I
t + iy
G(t + iy) G (iy) I
k-+ 00
which has the property that for arbitrary
y
s
> 0 and sufficiently large k
Rl (t, y) cg(;:{) > exp ( - 2r�1+8) provided that t 2 +y 2 r�. We consider next f2 {z) = exp [H(z)] and write (7. 3 . 3 8) R2 (t, y) = I exp [H(t + iy) -H (iy)] I so that i f(t y ) + (7.3 . 39) R(t, y) f(iy) = R1 (t, y) R2 (t, y).
(7. 3 . 3 7)
=
=
=
I
We give an indirect proof for theorem 7.3 .4 and assume therefore that f(z) is an entire characteristic function of order p > 2 and suppose that the exponent of convergence p1 of the zeros off (z) is less than p, Pi < p. We again apply theorem 7 . 1 .2 and see that necessarily � 1 for all real and
(7.3.40) R(t, y) t y.
223
ANALY TIC CHARACTERISTIC FUNCTIONS
p1 p
p 1n, H(O) �1 (z) = H(z) = R2 (t, y) = exp [A 1 (t, y)] .
< Since we have = where m is the degree of the polynomial = 0 (since f( O) = and use the notation of We see also that the preceding proof and write
H(z).
1)
� (rxv v=l m
v
+ if3 )z v
so that (7.3 .41 ) We see then from (7.3 . 37), (7.3 .39) and (7 .3 .41) that there exists an infinite sequence of indefinitely increasing positive numbers such that for an arbitrary s 0 > exp [ - 2 �1 + e + A1 (7.3 .42) provided that k is sufficiently large and that = We next define an infinite sequence of points in the z-plane. In order to be able to apply lemma 7.3 .3A we subject these points to the following restrictions : (i) = (ii) 1 1 = 0 (iii) if m 3 or m = 3 while ?' a = f3 a = 0, then (iv) if m = 3 and ?'a = f3a =I= 0, then { - sgn ?'a) 0. From (i) and (ii) it is seen that all these points are located in the same We deduce from lemma 7.3 .3A quadrant and that I I = that (7.3 .43) = A + o(1)] as k -+ oo where 0. We denote by Ql = and obtain from (7.3 .42) and (7.3 .43 ) as k � oo. exp { - 2 r�1 + e + Qlr� ) Since by assumption = m > we can choose the arbitrary positive quantity s so that s < m . We conclude from the last inequality that exp {Qlr� as /� � oo. Since Q( 0 we can determine k so large that This, however, contradicts (7.3 .40 ) and we see therefore that f(z) cannot be a charac teristic function and have thus completed the proof of theorem 7.3 .4. In conclusion we mention, without proof, another theorem of this type.
rk
> R(t, y)
(t, y)]2, 2 t +y r�. (tk, yk)
t1c Yk V�m t1c + iyk rk >
yk > yk >
Yk rlc/ v(1 + �m) · 1 (tk, y1,) .Aj Y1c Jm [ 1 A> A(1 +�m)-m1 2 [1 + o{ 1 ] } R(tk, yk) > p p1 , p1 + R(tk, yk) > [1 + o(1 )] } > R(tk, yk) > 1.
Theorem 7.3 .5.
Let
Pm (t) = � be a polynomial of degree The function f(t) = exp [lt1 (eit _ 1) +A2 (e - it _ 1) + Pm (t)] i f , and onl is haracter tic function i f , lt 0, lt 2 0, y 1 P1(t) = (it)-a2t 2 where a1 and a2 are real and a2 0. m
m.
a
c
a1
is
v=O
Cv l
v
�
�
�
m
� 2
and if
224
CHARACTERISTI C FUNCTIONS
This theorem contains again as a special case the theorem of Marcin kiewicz (corollary to theorem 7.3 .3). Marcinkiewicz ' s theorem is obtained by putting = = 0. For the proof, which is similar to the demonstra tion of theorem 7.3 .3 , the reader is referred to Lukacs ( 1 958). Several authors have discussed related necessary conditions for entire or meromorphic functions to be characteristic functions. These conditions can all be considered to be extensions of Marcinkiewicz ' s theorem. I . F. Christensen ( 1962) studied functions of the form
A1 A2
j(t) = Kn g(t) en [Pm (t)] ,
g(t)
where is a characteristic function subject to certain restrictions. R . Cairoli (1 964 ) investigated similar problems for meromorphic functions of finite order. H. D . Miller (1 967) studied entire functions of the form or {exp }, where and are entire functions {exp while is a polynomial. The method in all these cases is similar to that used in proving theorem 7. 3 .2 ; the principal tool is the ridge property (7 . 3 .44) 1� which is valid for all y if is an entire characteristic function. Far-reaching generalizations of Marcinkiewicz ' s theorem were obtained by I . V. Ostrovskii (1963 ). His work is based on a careful study of entire functions which belong to families characterized by the following in equalities : (7.3 .4Sa) � < oo) ( - oo < (7.3 .4Sb) � ( - oo < , y < co ) It is easily seen that the class described by (7 .3 .4Sb) contains the class described by (7.3 .45a) which in turn is wider than the family of ridge functions. The basic results of Ostrovskii ' s paper are theorems on entire functions belonging to these classes. These theorems are interesting on account of their applicability to the theory of characteristic functions. The reasoning which yields these results on entire functions is tedious, and the discussion would exceed the scope of this monograph. We therefore list, as a lemma, only one of Ostrovskii ' s results and also indicate its application .
g(t) f
P ( t)
[P(t)]} f
[P(t)]
g(t) f(t)
l f 0. We wish to avoid the discussion of trivial cases and suppose therefore that ¢= 1 . Let - rx < Im < {3, ( ex > 0, f3 > 0), be the strip of regularity of we first show that necessarily � min rx, {3) . We give an indirect proof and assume that < min ( ex , {3) . The points z2 = are then in the interior of the strip of regularity of and i t follows from theorem 7 . 1 .4 that (ir; ) +
f(t).
f(z)
(n
f(z)
f(z)
- in f ( 7 .4. 2) ! (O)
"�''
n
=
,
• ··
f ( i ) n "":i"·· · .
.
in
n
(z)
f(z);
min ( ex, {3). But then at least one of the inequalities n > ex or 'YJ > (3 holds. If n > ex [respectively n > {3] then (n - ex) [respectively - (n - {3)] is a singular point off(z) located in the upper [respectively lower] half-plane. Therefore n - ex � {3 , and we have established the following result :
f(O)
f( - i
fi
f
i f( - i
i
i
analytic characteristic function has a Theorem 7.4.1. If a non-constant purely imaginary period = in (n > 0), then this period is at least equal to the width of the strip of regularity off(z), that is I I = n � ex + {3. We consider next the case where f(z) has a complex period = � + ir . w
w
w
J
The case � = 0 (purely imaginary period) has just been treated, so that we may assume � =1= 0. Using (7. 1 . 4) and the assumption that w is a period of (z), we conclude easily that w and - w are also periods. Therefore 2� and 2ni are also periods of (z) so that f(2�) = 1 . ( 7.4.3) We conclude then from theorem 2. 1 .4 thatf(z) is the characteristic function of a lattice distribution whose lattice points are the points where 1 - cos 2�x vanishes. Therefore (z) is given by
f
f
f
(7.4.4)
f(z) =
i s
s = oo
� P s exp ( zn /�)
s = - oo
where (7 .4.5)
P s � 0,
f
00
� Ps = 1 . 8 = '-
00
If n = 0, then (z) is simply periodic and has a real period � ' so that = f(�) = 1 , and we see by the same argument that it can be written as
f(O)
(7.4. 6 )
f(z) =
00
� s = - oo
is
P s exp (2n z /�)
where the P s satisfy again (7.4.5). If n =I= 0 thenf(z) is given by (7.4.4) and is a doubly periodic function which necessarily has a real and also a purely imaginary period. We summarize this in the following manner :
Theorem 7.4.2. An analytic characteristic function which is single-valued and simply periodic has either a real or a purely imaginary period. The period is real if, and only if, the characteristicfunction belongs to a lattice distribution which has the origin as lattice point. Let f ( z) be an entire characteristic function which does not reduce to a a
constant and assume that it is periodic . From theorem 7 . 4. 1
we
see that it
227
ANALYTIC CHARACTERISTIC FUNCTIONS
cannot be doubly periodic, and we can conclude that it must have a real period and have the form
(7.4.6). Theorem 7 .4.3 . If a characteristic function is an entire periodic function then it is necessarily the characteristic function of lattice distribution which has the origin as a lattice point. a
It is easy to give examples of analytic characteristic functions which are periodic. We mention the Poisson distribution whose characteristic function has the real period 2n ; the distribution with frequency function = cosh (nx/2) ] - 1 has the characteristic function f(z) = 1 /(cosh z) which is regular in the strip Im (z) < n/2 and which has the purely imaginary period 2ni. A doubly periodic characteristic function was constructed by M . Girault ( 1 955) who showed that the elliptic function
[2
p(x)
I
I
f(z) =
2n - 1 1 + k 2n - I eiz 1 -k rroo 1 + k 2n - 1 1 - k2n -l eiz oo
..
is a characteristic function. This function has the real period 2n , the purely imaginary period log and the strip of regularity Im (z) < ! log k j . 7 .4.2 In conclusion we remark that one could regard theorems also as conditions which a single-valued, periodic analytic and function must satisfy in order to be a characteristic function.
7.4.3
4i k
I
I 7.4.1,
7.5
Analytic characteristic functions as solutions of certain differential equations Regression problems lead sometimes to a differential equation for the characteristic function. After all solutions of this equation are found, one has to determine those which can be characteristic functions. This is often the most difficult part of the problem and it is therefore desirable to find general properties of characteristic functions which satisfy certain differ ential equations. In the present section we discuss a result due to A. A. Zinger and Yu. V. Linnik, which is of great theoretical interest. We write for the derivative of order s of and consider the differential equation jn j ... l () . + l i(j f( l) ( ( = + .5 ) � Aj ...jn ' l'he Aj1... jn are real constants \vhile the sum is here taken over all non negative integers j1 , j , , In which satisfy the condition jt +j2 + . . - + In � = ( js � 0 ;
f<s> (t)
f(t) t) . . . fUn t) c [f t ] n
(7 .1
(7. 5 .1a)
2
•
•
•
m
s 1, . . . , n).
W c ass tune that at least one coefficient with j1 +j2 + . . . +In = rn is different from zero and denote by 'Ill the ord(!r of this d ifferential e qu ati o n .
228
CHARACTERISTIC FUNCTIONS
We adjoin to the differential equation (7.5 . 1 ) the polynomial
A (xl'
(7.5.2)
1
•
.
n s
. . . ' Xn) = -n , (s�o�....sn> �A X . . Jl···J
31 1
•
•
•
xs' nn. .
The first summation is here to be extended over all permutations of the numbers ( 1 , 2, . . . , n) ; the second summation over satisfying (7.5 . 1 a) . a ll integers The differential equation (7 .5 . 1 ) is said to b e positive definite if its adjoint polynomial (7.5 .2) is non-negative. We can now state the result of A . A . Zinger and Yu. V. Linnik.
{s1 , s2 , , jsn) ,jn 1, •
•
•
•
•
•
Theorem 7.5 . 1 . Suppose that thefunctionf(t) is, in a certain neighbourhood of the origin, a solution of the positive definite differential equation (7 .5 . 1 ) and assume that m n - 1 . If the solution f(t) is a characteristic function then it is necessarily an entire function. ;>,:
We state first a lemma , which uses only some of the assumptions of theorem 7.5 . 1 and which therefore yields less information concerning the solutions of (7 .5 . 1 .
) Lemma 7.5 . 1 . Suppose that the characteristic function f(t) is, in a certain neighbourhood of the origin, a solution of the positive definite equation (7 .5 ). Then f(t) has derivatives of all orders at the origin. Lemma 7.5 . 1 is certainly true if the distribution function F (x) of f(t) .1
is a finite distribution [see theorem 7.2.3] . We therefore assume in the following that for all x > 0 (7.5.3) F ( - x) + 1 - F (x) > 0. We remark that the assumptions of the lemma imply that can be differentiated at least times. Moreover , is necessarily an even number if is non-negative. Since is the characteristic function of F (x), we know that
m
A(x1 , x2 , , Xn) (7 .5 .4) J . . . , x..) (x1 + . . . + x.,)2P-m+ 2 exp [it(x1 + . . . + x )] dF (x ) • • • dF (xn) 00 00 J 00 • • • f 00 (x1 + . . . + x.,)2P -m+2 exp [it(x1 + . . . + x )] F(x1) • • • dF(xn) d or , putting t = 0, (7.5.6) s: <Xl . . . s: <Xl A(xl> . . . ' x..)(xl + . . . + x..)2P -m+ 2 dF (x1) • • • dF (xn) = f 00 • • • f : (x1 + . . . + x,.)2p-m+2 dF (x1) . . . dF (x,.) . 00 The differential equation (7 .5 . 1 ) has by assumption the order m so that the polynomial A(x1, • • • , xn) contains the mth power of at least one variable. It is evidently no restriction to assume that x1 is this variable ; one can then write (7.5 .7) A(x1, • • • , Xn) = A0 (x2, • • • , Xn)x�+ A 1 (x2, • • • , Xn)x� - 1 + . . . + Am (x2, . . . ' Xn) · Since A(x1, • • • , Xn ) is non-negative we conclude that A 0 { x 2 , • • • , x,.) is also a non-negative polynomial. It is always possible to find a bounded region 0'17,_1 in the ( 1) dimensional space Rn _1 of the variables (x 2 , • • • , Xn) such that tln _ , dF(x2)dF(x3) • • • dF(x..) = oc > 0 (7.5 .8a) while (7.5 .8b) min nO -1 A o (x2, Xa, • • • ' Xn) � > 0. This follows from (7.5 .3) an d the fact that the equation A0 {x2, • • • , Xn) = 0 determines an algebraic surface in Rn_1• We use here, and in the following, the symbols C1 , C 2, • • • to denote arbitrary positive constants. We see from (7 .5. 7) that it is possible to find a sufficiently large C 2 > 0 such that for I XI I > c2 and ( x 2 , . . . ' Xn) nn-1 the relations C3xr;: A (x , • • • , Xn) {I x + -1- • • • x.,, I I x1 I < 7 ·5 ·9) �
�
m
..
x
=c
x
1
..
x
c
00
n
cl
1
hol d .
•.
x2
+
�
E
� c4
-
230
CHARACTERI STIC FUNCTIONS
( x 1 , , n) Qn -1· (x2, I x l l c2 m A(x1 , x2, , n) {7 .5 . 1 1 ) f A(x1 , , ) (x1 . . . n)2P -m + 2 dF (x1) dF (xn) K Qn
of the n-dimensional be the set of all the points Let X space which satisfy the condition . . . ' X-n) E and (7.5 . 1 0) > Since is even and � 0 we conclude from (7.5 .6) that X •
•
nn
where
•
•
•
•
•
•
•
Xn
-� X
+
•
•
•
�
f ' J: ro (x1 . . . + x,.)2l> - m + 2 dF(x1) dF (x,.) is a (finite) positive constant. Substituting (7.5.9) into (7.5 . 1 1), we see that K
=
c
C3 qv- m + 2 f
00
•
•
+
•
•
xiv + 2 dF (x1) dF (x,.) C5 Jf
•
•
+ 2 ) dF(x x't_P ::; ; K. 1 J This inequality indicates that the moment of order 2p 2 of F (x) exists ; this is in contradiction with the assumption concerning p, so that the On
•
•
=
•
lx1l > Gil
+
indirect proof of lemma 7.5 . 1 is completed. We proceed now to prove theorem 7.5 . 1 . As a first step we show that is an analytic characteristic function. We need the following lemma.
f(t) Lemma 7.5 .2. Let G be a positive integer; then eG > GGjG ! . To prove the lemma we note that
eG
=
G i GG . > � G '. J. .t 3- = 0 00
m be two positive integersN ; according to lemma 7.5 .2 we have (2N m)2 +m < e2N + m m)! (2N + or 2N + m (7.5 . 1 2) < e. N 1/( ) ] + m 2 ! [(2N m) We again use the region Qn - t introduced in the proof of lemma 7.5 . 1 and write Xn l i dF (x 2) dF ) (x h; I 2 · n f It is then possible to find a positive number b such that (7.5 . 1 3 ) h; < b i b 0 (j 1 , 2, . . . , 2N ). We consider also the set of those points (x1 , x2 , , xn) of the n-dimen sional space which satisfy the relations (7. 5 . 1 4) I I c2 and ( n) nn
Let N and
+
+
=
nn - 1
X
+
X3
+
. . .
+
•
•
•
=
•
Xt
�
X 2 ' X 3'
•
•
•
'
X
E
•
•
-1 ·
This set is bounded, therefore there exists a positive constant C6 such that
231
ANALYTIC CHARACTERISTIC FUNCTIONS
(7.5.15) I = J
2N x x x . ) ( x J + + l n r;: l [ l ( · · 2 l 0 such that 1 (7.5.29) n -M < Yo
""' dF(x) JB /wu:=>"" dF1 (z)J dF2( y) .
We note that
>
>
238
CHARACTERISTIC FUNCTIONS
so that
I"' 00 emedF(x) [J B /1YY dF2 ( y)J [ I:+ : evzdF1 (z)J . �
The integral on the left of this inequality is finite and independent of Carrying out the necessary passages to the limit, we see that the integrals
a, b, A, B.
I "' oo evz dF1 (x) I oo oo e�'11 dF2 (y)
(8.1.7)
and
exist and are finite and that
(8.1.8) Here v is a real number such that - oc < v < {3, so that the integrals (8.1.7) exist for all such v. But then the integrals /1 (z) = I: oo ei""' dF1 (x) and /2 (z) = I: oo ei""' dF2(x) exist and are finite for all complex z su ch that - oc < (z) < {3, and we see that f1 (z) and f 2 (z) are analytic characteristic functions whose strip of regularity is at least the strip of f(z). Moreover equation ( 8.1.1), which holds for real t, is also valid (by analytic continuation) in the entire strip of regularity of f(z). We summarize this result as Theorem 8.1.1. Letf(z) be an analytic characteristic function which has the strip - lm (z) < f3 as its strip of regularity. Then any factor f1 (z) of f(z) is also an analytic characteristic function which is regular, at least in the strip of regularity off (z). We now turn back to inequality (8.1.8). There exist two real numbers a1 and a2 such that 0 < F2 {a1) while 1 > F2 {a2). Then 00I dF2 (x) ea"' [1 - F2 {a2)] f v > 0 J'" 00 e1YY dF2 (y) r·"' e""' dF2 (x) ea•• F2 (a1) v < 0. Let C - 1 = min [F2 (a1), 1 - F2 (a 2)] and a = max [I a 1 j, I a 2 j ] ; we then see that J �oo e..., dF1 (x) � C tfl"l I oo e,., dF (x). Corollary to theorem 8.1.1. Let f(z) be a decomposable analytic charac teristic function with st1�ip of regularity - oc < m ( ) < {J and suppose that Im
ex
1 and type 7: and suppose that f1 (z) is a factor off(z). If the order of f1 (z) is also then the type off1 (z) cannot exceed the type 7: �f f(z), 7:1 � 7:. �
p
1:1
p,
The statement of the corollary is obtained in the same way as the state tnent of the theorem , using the definition of the type given in Appendix D.
Re11zark 1. The statement of the corollary does not hold if either = or if < where p1 is the order of f1 (z). Remark 2.e (zLet f(z) be an entire characteristic function without zeros so that f(z) = cb > [4>(z) entire, z = t + iy] . Then every factor f1 (z) of f(z) is also an entire characteristic function without zeros and therefore has the form f1 (z) = ecb1 (z> where 4>1 (z) is an entire function. p1
1
p
p,
'Ve conclude this section by deriving a property of entire characteristic functions without zeros.
Theorem 8.1.3. Let f(z) be an entire characteristic function without zeros which has a factor f1 (z). The entire functions 4>(z) == log f(z) and 4>1 (z) = log f1 {z) then satisfy the relation M(r ; 4> 1) 6rM(r + 1 ; 4>) + Cr(r + 1), where C is a positive constant. For the proof of the theorem we need two lemmas. (*) Le1nnza 8.1.1. Let f(z) be a function which is regular in a region G, let point of G, and let � be the distance between z 0 0 == t 0 + iy 0 be anofinterior and the boundary G. Then -z ) d0+ i{J 0 • (8.1 . 9) f(z) = 1 I u(t0 + cos 0,y0+ s 0) e -+ ((zz-z0 ) �
z
2
n
n
2n
o
p
p
m •
p ew
p
w
( • > J.jet w = f + iTJ be a complex number nnd let f(w) be a function which cct·tnin region. We write then u(�, 'YJ) for the real part of f(w).
o
is
.
regular in
240
CHARACTERISTIC FUNCTIONS
Moreover, 2n 1 (8.1.9a) f '(z ) = J u(t0 + cos 0, Yo + s1n. 0) e- "'.0 dO. ere is an interior point of the region G; is a point in the interior of the Hcircle with centre z0 and radius such that I I < < � ' while flo = lm [f(zo)] . The representation (8.1. 9 ) is known as Schwarz ' s formula ; for its proof see Markushevich (1965) [vol. 2, p. 151 ] . If we differentiate (8.1. 9 ) with respect to z and put = z we obtain (8.1. 9a). Lemma 8.1.2. Letf(z) be an entire characteristic function [z = t + iy ; t, y real], then there exists a positive constant M = M1 , which depends on f but is independent of y, such that log f(iy) - M I y I · -yK1, where K1 = if'(O). According to theorem 7.3. 2 we have log f(iy ) The statement of the lemma follows from the fact that -y K1 K1 I I y I I so that in this case M = I K1 I · If K 1 = 0, M is an arbitrary if K1 0, positive number. We proceed to the proof of theorem 8.1.3 and write (S. l . l O) {uu(t,1 (t,yy) )==ReRe[4>[c/>(t1+(ti+y)]iy)] It follows easily from theorem 7 .1.2 that (8.1.1 1 ) 0 � u1 (0, y)- u1 (t, y) � u(O , y)-u(t, y) 2M(r; 4>), where r = I t+ iy I · Since f1 (z) is a factor of f(z) there exists an entire characteristic func tion f2 (z) without zeros such that f(z) = f1 (z) f2 (z). We write 4> 2 (z) = log /2 (z) and u 2 (t, y) = Re [4> 2 (t + iy) ] , so that (8.1.12) u1 (t, y) == u(t, y) - u2 (t, y). We conclude from lemma 8.1. 2 that there exist positive constants M1 and M2 such that log f1 (iy) - Mj I y I (j = 1, 2) and note that u1 (0, y) = log I f1 (iy) I == log f1 (iy). Hence (8.1.13) (0, y) - Mj I y I (j = 1, 2). It is also easily seen that (8.1.14) u(t, y) = log I f(t, y) I � M(r 4>), where r = (t 2 + y 2)1 12 • It follows from (8.1.12), (8.1.13) and (8.1.14) that - M1 I Y I � u1(0, y) = u( O , y )-u2 (0, y) � u( O , y ) + M I Y I 0
-
np
p
p
o
z0
z
p
z
z - z0
p
0
�
�
�
=I=
�
�
Uj
�
;
2
24 1
FACTORIZATION OF ANALYTIC CHARACTERISTIC FUNCTIONS
or
-M1 I Y I � u1 (0, y) � M( y; 4>)+M2 I Y I · Therefore there exists a positive constant C such that (8.1.15) I u1 (0, y) I � M(y; 4>) + C I y I · Clearly I u1 (t, y) I � I u1 (0, y) I + I u1 (0, y) -u1 (t, Y) I or, using (8.1.11) and (8.1.15), l u1 '(t, y) l � M( y; 4>)+ C i y i + 2M(r ; 4>). Since I y I � r we have (8.1.16) l u1 (t, y) I � 3M(r ; 4>) + C I y I · We apply formula (8.1.9a) of lemma 8.1.1 to 4>1 (z). Since 4>1{z) is an entire function we may put p 1 and we write also z, t and y instead of z0, t 0 and y0, respectively. We obtain cP� (z) � J 2" u 1 (t + cos (), y + sin 0) e - w dO. It follows from (8.1.16) that 1 4>� (z) l � 6M(r + 1 ; 4>) +2C (r + 1). ==
=
1'l
0
Since
we see that
1 4>1{z) l 6rM (r + 1 ; 4>) + 2Cr(r + 1 ). This is the estimate given in theorem 8.1.3. Corollary 1 to theorenz 8.1.3. Let f(z) be an entire characteristic function without zeros which has a factor f1 (z) and write 4>(z) l g f(z) , 4>1 (z) log f1 (z). The order p 1 of 4>1 (z) cannot exceed the order p of 4>(z). Moreover , if p1 p then the type of 4>1 (z) cannot exceed the type of 4>(z). �
==
o
=
=-
The statement of the corollary follows immediately from the theorem and the definitions of order and type of an entire function given in Appen dix D.
Remark.
8. 1.3 (0, y) 1 (0, y)-u1 (t, y) (8.1.11 ) (8.1.15). y) (0 , y) - y) l u1 (0 ,y) l B(y)
The estimate of theorem can sometimes be improved , and u better namely if it is possible to find for u1 bounds than those of formulae and Suppose that we have (8 . 1 .1 1 a) 0 � u1 u1 (t, � A( t ,
(8. 1 .15 a)
�
242
CHARACTERISTIC FUNCTIONS
A(t, y)
B(y)
where are non-decreasing functions. Repeating the pre and vious argument we get � ttl and � We give an example which we shall use in the next section. Let exp where is real, while � and � We suppose that f z) admits the decomposition,
I (t, y) I A(t, y) + B(y) ( 8.1.17) 14>I (z) l 2 l z i A(t + 1 , y + 1) + 2 l z i B(y+ 1) (z t+iy) . f(z) {A(eiz _ 1) + ipz-yz2}, y 0 A 0. ( p f(z) f1 (z)f2 (z). The function f(z) is an entire characteristic function without zeros ; we write again u(t, y) Re [log f(t + iy )] Re [4>(t + iy )] and use analogous notations for the factors f1 (z) and f2 (z). Then f(i t y ) (8.1.18) u(O ,y)-u(t,y) = log f(t + iy) = 2A[11 sin 2 2 + yt2, so that we see from (8.1.11 ) that (8.1 .19) A(t, y) 2A elvl +y t 2 . According to lemma 8.1.2 there exist two positive constants M1 and M2 such that =
=
=
=
=
=
and we see that . . a According to our assumptions we have
(8 1 20 )
so that (8. 1 .20b) Hence so that
1>1 (iy) log f(iy)- 4>2 (iy) 1U (0, y) � A(e- v- 1)- ttY + YY 2 + M2 1 Y I · I Ut (0, y)] :( Aelvl + yy 2 + O( I Y 1) , =
(8.1. 2 1) We see therefore from (8.1.17 ), (8.1.19) and (8.1.21) that (8.1.22) l 4>1 (z) l O { l z l exp [ I Im (z) I J + I z l 3 } ( l z l =
and have obtained the following result :
� oo
)
Corollary 2 to theorem 8.1.3. Let f(z) be the characteristic function of the convolution of a normal and a Poisson distribution, f(z) = exp [A(eiz _ 1) + ittz-y 2] . z
243
FACTORIZATION OF ANALYTIC CHARACTERISTIC FUNCTIONS
Iff1 (z) = exp [4>1 (z)] is a factor off(z), then 4>1 (z) = O { l z l exp [ j lm (z) IJ + I z J 3 } as I z I -+ oo .
8.2 Factorization of certain entire characteristic functions Certain entire characteristic functions have interesting factorization properties. We next prove an important theorem concerning the decom position of the normal distribution ; this theorem was first conjectured by P. Levy and somewhat later proved by H. Cramer.
Theorem 8.2.12 2(Cramer's theorem). The characteristic function f(t) = exp [i,ut- a t /2] of the nornzal distribution has on�y 2normal factors. Mthenoreover, if f(t) =df1 (t)f2 (t) with2 • fi (t) = exp [itti t-a�t /2] (j = 1, 2), tt 1 + tt 2 = tt an ai +a� = a The function f(t) is an entire function without zeros ; it follows then from theorem 8 . 1 .2 that the same is true for its factors and that the order of these factors cannot exceed 2. Therefore f1 (z) has the form f1 (z) exp [g1 (z)] and it follows from Hadamard ' s factorization theorem that g1 ( ) is a polynomial of degree not exceeding 2. Let for real argument t, g1 (t) = a0 + a1t+a2t 2 ; since f(O) = 1 we see that a0 = 0. From the rela tion g1 { - t) = g 1 (t) we conclude that a1 = itt 1 is purely imaginary and that Since a characteristic function is bounded for real values of its aargument 2 is real. we deduce finally from l f1 (t) j = exp [a2 t 2] that a 2 � 0 and set a2 = - lai. Thus f1 (t) = exp [itt 1 t-lai t 2] is the characteristic function of a normal distribution. The same argument applies to f2 ( t) while the relation between the parameters off(t) and those of its factors is established by elementary reasoning. =
z
We discussed in Section 6.2, without giving any examples, characteristic functions which have no indecomposable factors. Cramer ' s theorem shows that the characteristic function of the normal distribution belongs to the class of characteristic functions without indecomposable factors. Our next theorem indicates that the characteristic function of the Poisson distribution also belongs to this class. The following factorization theorem was derived by D. A. Raikov and is in some respects similar to Cramer ' s theorem.
's theorem). The characteristic function f(t) Theorem 8.2.2 (Raikov Poissonian factors. has onl =Morexp [A(eifit_ 1)] of the Poisson distribution y eover f(t) = f1 (t)f2 (t) with f1 (t) = exp [Ai (eit - 1)] (j = 1 , 2) then A1 A2 = A. -1-
To prove the theorem
(8.2. 1 )
.f ( /)
�=
we
suppose that cxp [A ( £lu - 1 )] .f1 ( t).f.,. ( t) ==
244
CHARACTERISTIC FUNCTIONS
is decomposed into two non-degenerate factors. Since the convolution of a discrete and a continuous distribution is always continuous , we see that /1 ( t) and /2 ( t) are necessarily characteristic functions of discrete distri butions. The Poisson distribution f(t) has its discontinuity points at the non-negative integers ; it is then no restriction to assume that the discon tinuity points of /1 ( t) and /2 ( t) are also non-negative integers. Then /1 (t) = � av eitv and /2 { t) = � bv eitv with av � bv � v=O v=O 00 _lV and where f(t) = e- J. � , e"'tv . v=O v=O V . v=O Since f(t) is an entire function without zeros, the same is also true for /1 ( t) and /2 ( t), so that these series also converge for arbitrary complex values of the argument. We now introduce a new variable w = eit ; this transforms the characteristic functions /1 ( t), /2 ( t) and f ( t) into the gener ating functions g1 (w), g 2 (w) and g(w) respectively. Here � i. g(w) = e- � , w v , g1 (w) = � av wv , g 2 (w) = � bv wv v =O V • v=O v=O and g(w) = g1 (w) g2 (w). The coefficients of these power series satisfy the equation 00
0,
00
0,
•
-
00
00
00
-
(8. 2 . 2) and it follows from the non-negativity of the av and b v and from the relation that
_Av 1 i. av � b0- e - , . v. Since g(w) = exp [.A(w - 1 )] is an entire function, we conclude from (8.2.3) that g1 w) is also an entire function, and we see that for real t g1 (t) � b(; 1 g(t). It is also easy to verify that M(r ; g) � b 0 M (r, g1) so that the order of g1 (w) cannot exceed the order of g(w) . The function g(w) is an entire function of order 1 without zeros ; therefore g1 (w) has the same property. We conclude from Hadamard's factorization theorem that g 1 {w) has exactly the order 1 . Since g1 (1) = we see that g 1 (w) = exp [.A1 (w - 1 )] so that /1 (t) = exp [.A1 ( eit 1)] is the characteristic func tion of a Poisson distribution. A similar argument applies to /2 ( t) and it is A1 + A2 = A. A2 � easily seen that A 1 �
( 8.2.3)
-
"
(
1
_
0,
0,
245
FACTORIZATION OF ANALY TIC CHARACTERISTIC FUNCTIONS
The following corollary follows almost immediately from Raikov ' s theorem.
Corollary to theorem 8.2.2. A Poisson-type distribution has only Poisson type factors. One can summarize theorems 8. 2 .1 and 8.2. 2 by introducing the follow
ing definition. A family of characteristic functions (or distribution func tions) is said to be factor-closed if the factors of every element of the family belong necessarily !O the family. The preceding results mean that the nor mal family, as well . as the family of Poisson type distributions, is factor closed. H. Teicher showed that a family which contains the binomial distributions is factor-closed. For the binomial distributions this fact was In this connection we mention an already noted by N. A. Sapogov interesting result which describes another family of characteristic functions which is factor-closed. Yu. V. Linnik derived the following generalization of the theorems of Cramer and Raikov.
(1954)
(1951).
( 1957)
Theorem 8.2.3. Let = exp {A(eit _ 1 ) + i,ut- � a 2 t 2 } f(t) (,u real, a 2 0, A � 0) be the characteristic function of the convolution of a normal and of a Poisson distribution. Suppose that f(t) has the decomposition f(t) = f1 (t)f2 (t) . Then f3 (t) = exp {Ai (eit - 1) + itti t- !a� t 2 } (j == 1 , 2) where A = A1 + A2 and a2 = ai + a�. We note that theorems 8. 2 .1 and 8.2.2 can be obtained from this result as particular cases. However, the proof of theorem 8. 2 .3 requires more powerful analytical tools than theorems 8.2.1 and 8.2. 2 . This is explained by the fact that theorem 8.2.1 deals with an entire function of finite order �
while theorem 8.2.2 treats the characteristic function of a lattice distri bution. Under the assumptions of theorem 8.2. 3 both these advantages are lost and the proof becomes much more complicated. For the proof of theorem we need certain results from the theory of analytic functions which we state as lemmas.
8.2.3 Lemma 8.2.1. Let f(z) be a function which is regular in the angle == {z : 0 z I < oo, ex arg (z) � f3 } and which satisfies the following I conditions: where < n/({3 - ) (i) l f (z) I M1 exp {I z I P) for z (ii) l f (z) I � M on the lines z == ei cx and z == x e if1 forming the boundary of Then l f(z) I M for all z :(
!!)
:(
:(
�
E !?!)
!!) .
·�---- --------
,-fhnt is
n
x
E
p
!!).
d istribution w i th characteris tic fu n c ti on of the form
f( t) =
cxp
[tip. ··I· A(e'l.ll1 - 1 )].
ex .
4
2 6
CIIARACTERISTIC FUNCTIONS
Lemma 8 .2. 1 is a special case of the Phragmen- Lindelof theorem ; for its proof we refer the reader to Titchmarsh or to Markushe [p. vich [vol. 2, p . 214] .
(1939) 176] (1965) Lemma 8.2.2. Let f(z) be an entire periodic function with period T, such that the inequality l f(z) I K ea lzl (K and a are real and positive constants) holds. Then 2 i f(z) = ±T ck exp { ; zk} is a trigonometric polynomial with = [a I T l /2n] . Lemma 8 .2.2 is a consequence of the theorem [see Markushevich (1967), vol. 3, p. 143 ] which states that a non-constant, periodic entire function of exponential type is necessarily a trigonometric polynomial. :(
k
7:
We proceed to prove theorem 8 .2.3 and suppose that the characteristic function (8.2.4) admits a decomposition
(8.2.5) f(t) = /1 (t)/2 ( t).
We see from theorem 8 . 1 .2 and the fact that f(t) is an entire character istic function without zeros that are entire characteristic and /2 functions without zeros. Therefore = exp [ 1 (j = 1 , 2) where the are entire functions which are real for = = ( We see oo, real) and have the property that from corollary 2 to theorem 8 . 1 .3 that = (8.2.6) I exp [ I Im I J + I 1 3 } (I I -+ oo). Up to this point we have considered = Re [4> 1 (z)] as a function of the real variables and y . For the completion of the proof it is necessary and to continue to y) into the complex plane. It will be convenient to introduce the function = (8. 2.7) We note that is an entire function which is real for real Therefore admits an expansion
fi (z)
(z) 4>i - oo < y < y 4>1 (z) O { l z t u1 (t, fix y g(z) 4>1 ( -iz). g(z) g(z) a1c
(t)
/1 (t) 4> (z)]
4>1 (0) 0.
(z)
z u1 (t, y)
z
z.
g(z) =
00
�
k=O
ak zk
z t iy (t, y real) then [g(z)] [g(t i ] �[g(t iy) + g(t - iy)]. [4>1 { -iz)] [4>1 (y- it)] = u1 (y, -t)
where the coefficients are real. Let = + Re = Re + + y) = On the other hand, we see from (8.2. 7) that = Re Re [g(z)] = Re
z iy
FACTORIZATION OF ANALYTIC CHARACTERI STIC FUNCTIONS
247
so that (8.2.8) u1 , 'fhe right-hand side of equation (8 .2.8) is, for fixed an entire function of t and can be continued into the complex plane. In the following we write w) if we consider also complex values of the second variable. Since u1 w) 1 1( w), we see from (8.2.8) and (8.2.6) that <w >l (8. 2 .9 ) w � oo . w 1 3) w) 0( 1 w u1 We now introduce the function K (w) u 1 (0, w) - u1 ( 2n , w) . Since u1 (0, w) and u1 (2n, w) are entire functions, we see that K (w) is an entire function. It follows from (8. 1 . 1 1) and (8. 1 . 1 8) that 0 and w � oo, (8.2. 10a) K (w) 0( 1 ) if Im (w) while one sees from (8.2.9) that (8 .2. 10b) K (w) 0 and w -+ 0( 1 w 1 3) if Re (w) Moreover , one has for all w (8.2. 10c) K (w) as w � oo. O[exp ( I w We use the last three estimates to prove the following statement.
(y -t) = � fg(t + iy)+g(t - iy)] .
(y,
y,
g(w+iy) = � (y- i g(w-iy) == � -y- i (y, = I eiRe + I {I I == = = =
I I
=
= I I
1 3 12)]
Lemma 8 .2.3 . The fu ct
)
I I oo.
reduces to a constant. We consider the function {}(w) == K (w)(w + 1 ) - 3 . n
ion K (w)
This function is analytic in the half-plane Re (w) � 0 ; in view of the estimates (8.2. 10a), ( 8.2. 1 0b) and (8.2.10c) , it satisfies the conditions of lemma 8.2. 1 in each of the angles
-�
�
arg (w) � 0 and 0 � arg (w) �
�-
We conclude from lemma 8.2. 1 that for Re (w) � 0, (8.2 . 1 1 ) K (w) 0( 1 w 1 3) as w -+ oo. We use the function {}1 j w l K (w)(w - 1) -3 , which is analytic in the half-plane Re (w) � 0, to show in the same way that (8 .2. 1 1 ) is valid also for Re (w) � 0. Therefore the entire function J( ( w) satisfies (8.2. 1 1) for all w, so that it is necessarily a polynomial of degree not exceeding 3. We conclude from the estimate (8.2. 10a) that J( ( w ) is necessarily a constant.
==
I I
=
( • > with J
p =
3 /2, {3 - IX
=z 1r
/2.
248
CHARACTERISTIC FUNCTION S
We are now ready to complete the proof of theorem 8.2. 3 . It follows from the definition of the function and from that = is a constant, we see Since, according to lemma the function that satisfies the relation ) g( z (z) = (z ) where is a constant. V\Te put z (8 .2. 12) g1 (z) == g(z) (z 2n ) .
K(w) (8.2.8) -2K( - w) g(w + 2ni)+ g(w -2ni)- 2g(w). 8.2.5, K(w) g(z) g +2ni + - 2ni -2g c, c c -g - i - 2nt and see from the preceding equation that g1 (z) is periodic with period 2ni. Moreover we see from (8.2.6) that oo . g1 (z) = O [exp (3 1 z l /2)] as I z I The function g(z) satisfies the conditions of lemma 8.2.2 (with T = 2ni, a = 3 /2 == t'), and applying it we get g1 (z) == A0+A1 ez + A 2 e -z, where A0, A 1 , A 2 are constants. We see from (8.2.6) that g(z) == 0( 1 z j 3) as Re (z) -+ - oo ; the same is therefore true for g1 (z), so that A 2 = 0 and (8.2.13) g1 (z) == A0+A1 ez. In view of (8.2. 1 2) and (8.2.13) we have g(z)-g(z- 2ni) == B0+ B 1 z+B 2 ez with B 0, B 1 and B 2 constant. We put B B0+iB B 1 2 1 2 g2 (z) = g(z) z - -. ze. . . z-2nz 4-nz 2nz Repeating the reasoning \vhich led to (8. 2 .13), we see that g 2 (z) == C o + Cl ez, where C0 and C1 are constants. Using the definition of g2 (z) we conclude that ( 8.2.14) g(z) == D 0+ D 1 z+ D 2 z 2 + D 3 e + D 4 zez where the coefficients Di (j == 0, 1 , 2, 3, 4) are constants. These constants are real, since g(z) is real for real z. We see from (8. 2 . 7 ) that g(O) = 0; therefore, D o == - Da. We put z == y + it and separate the real and imaginary parts in ( 8.2.14) and obtain u1 (y, t) D1y+ D2 (y 2 - t 2)+ D3 (ev cos t - l ) + D4 e11(y cos t - t sin t). It follows from the estimate (8.1. . 20b) that D4 == 0. Therefore ul (y, O)-u1(y , t) = D2 t 2 + 2D3 eY sin 2 �. -+
=
249
FACTORIZATION OF ANALY TIC CHARACTERISTIC FUNCTIONS
t.
t=
This expression must be non-negative for all real y and If we put n and let y tend to oo we see that and if we put t n and let y tend to oo we see that D 2 Therefore
D3 � 0, + = � 0. g(z) = D1 z + D2 z 2 + D3 (ez- 1) (D2 � 0, D3 � 0). If we write D 1 ft 1, D 2 = y = �ai, D3 = A 1 we see that �1 (z) g( -iz) has the form �1 (z) = itt 1 - ai Z 2/2 + A 1 (eiz _ 1 ) ==
=
z
so that the theorem is proved. Some of the factorization theorems for analytic characteristic functions admit interesting generalizations which we discuss in Chapter The results of Raikov's theorem can also be extended in another direc tion. P. Levy b and D. A. Raikov studied the multiplicative structure of finite convolutions of Poisson type distributions and obtained a number of interesting results. We now introduce certain notations which will be used in formulating these results. Let be a real constant ; we denote by
(1937 )
(1938)
A�0 (8.2.15) F(x; A) = e- A � klA s(x- k) oo
9.
k
k o
A. (8.2.15); (t)
the distribution function of the Poisson distribution with parameter We write therefore for distributions of the type of clearly the characteristic function of Poisson type distributions is f == exp where a > and y are real numbers. Let a 1 a an be n positive numbers ; we write A{a1, a2 , , an) for the set of real numbers which can be represented in the form
F [(x-y)/a ; A] [iyt+A(eita _ 1 )] A � 0, 0 < 2< ... < + gn an, g1 a1 + g 2 a 2 + where the g 1 7 g 2 , , gn are arbitrary non-negative integers such that g1 + g 2 + . . . + gn > 0. The set A(a 1 , a 2 , . . . , an) has no finite accumulation point ; it is therefore possible to arrange its elements in an increasing •
•
•
· · ·
•
sequence
•
•
A1
==
a1
< A 2 < . . . < An . . . .
We say that the n numbers a 1 , a 2 , . . . , an are rationally independent if no relation
8.2.16) r1 a1 + r2 a2 + . . . + ron an = 0 holds where the r1, r 2 , . . . , rn are rational numbers such that l r1 l + l r2 ! + . - + l rn l > 0. other words linear relation (8.2.16) with rational coefficients between
(
In the
ah
a
·
. . . , lT,,.� can only hold if all the coeffi cients are zero. Wc now state l{aikov's resu lts.
250 Theorem 8 .2.4. Let y1, y2 , • • • , Yn be n arbitrary real numbers, and let A1, A2 , • • • , An be n non-negative numbers while 1, 2, • • • , are n positive numbers. The characteristic function of the distribution F(x �:n ; An) F(x) F(x �1Y1 ; A1) F(x �2Y2 ; A2) then has only factors of the form 1 )] exp [iyt + � ( where the Ak are the elements of A( 1, • • • n) and where y and are real numbers. We note that the factors of F(x) are not necessarily convolutions of CHARACTERI STIC FUNCTIONS
a
*
*
=
Ak < an
a
. . .
an
a
*
iA kt cxAk e ,
ocAk
a
Poisson type distributions since the coefficient ocA k may be negative. We will give later an example of such a characteristic function.
Theorem 8.2.5 . Let , be n rationally independent positive • • • numbers. The distribution x x x y 1 2 �:n ; An) ( ) ( ( ) �: � A A F F(x) F 1 ; 1 F ; 2 then has only factors of the same form, namely � � � x 1 X X 2 ) ( ) ( ( F1 (X) F 1 ; #1 F 2 ; #2 • • • F n ; fln) where �i �. 0 while O � fli � Aj(j 1 , ) Paul Levy ( 1 93 8b) has shown that theorems 8.2.3 and 8.2.4 are valid even if the numbers a 1 , a 2 , • • • , have arbitrary signs. Theorem 8.2.6. Let a1, a2 , • • • , be n positive numbers which satisfy the condition The distribution x x x y 1 2 * ) ( ( ) ( � F �:n ; An) F(x) F 1 ; A1 F �: ; A2 has only components of the same form, namely !5 � � x x 1 X 2 F1 (X) F( 0'1 ; #1) F ( 0'2 ; #2) • • • F( n ; ftn) where �i 0 while 0 � fli � Aj(j 1, 2, . . . , n). a1 , a2 ,
an
*
=
=
a
*
* . . .
a
=
*
*
*
an
. . . , n .
an
an
*
=
=
�
*
. . .
*
*
*
O'n
=
For the proof of the last three theorems the reader is referred to the paper by Raikov ( 1 93 8).(t)
(t) We prove in the next chapter two theorems [theorems 9.4. 2 and 9.4.4] which generalizations of Raikov's theorems 8 . 2 . 6 and 8 . 2. 5 respectively.
are
251
FACTORIZATION OF ANALYTIC CHARACTERISTIC FUNCTIONS
(1 37 )
In his remarkable paper P. Levy 9 b studied convolutions of Poisson type distributions of a somewhat more specialized character. He considered real polynomials and the entire functions exp If the coefficients of are all non-negative, then the coefficients in the Taylor series expansion of exp about the origin are also non-negative so that is a moment generating function. How exp ever, can be a generating function even if has negative co efficients. P. Levy derived a necessary and sufficient condition which the polynomial ::m : ust satisfy in order that exp should be a probability generating function. The generating functions studied by P. Levy belong to distributions of the form
P(x) P(x) P(x)] [ [P(et) -P(1)]
M(t) M(t) P(x) =
[P(x)].
P(x) [P(x)-P(1)]
F1 (;1 ; A1) * F2 (:2 ; A2) * . . . * Fn (:} An) where the positive numbers a 1 , a 2 , , an are all integers. We conclude this section with the discussion of an example which was •
•
•
studied by D. A. Raikov as well as by P. Levy. We consider the polynomial
(8.2.17) P(x) 1 +ax-f3x 2 + cx3 +dx4 of degree four. The numbers a, {3, c , d are assumed to be positive. We compute [P(x)] 2 and [P(x)] 3 and see that it is possible to determine the coefficients a, {3 , d in such a way that [P(x)] 2 and [P(x)] 3 have no k negative coefficients.(t) Then [P(x)] 2k P(x)]2} and [P(x)] 2k + 1 {[ [P(x)]3 {[P(x)] 2} k - t also have non-negative coefficients for k 1. We form (8.2.18) exp [P(x)] � [P��W and see that under these conditions only the quadratic term in exp [P(x)] can have a negative coefficient. The coefficient of the quadratic term of exp [P(x)] is easily determined ; it is [(a 2/2) -{J]e. If we suppose that (8.2.19) then exp [P(x)] has only non-negative coefficients. The function exp [P(x)-P(1)] is then a generating function, so that (8.2.20) g(t) exp {a eit -f3 e2it + c e3it + d e4it -a-c- d + {3 } a characteristic function. The function (8. 2 .20) cannot be an infinitely =
c,
=
=
;::::
=
=
0
1·
=
is
divisible characteristic function. To show this we note that the coeffi cients of the linear and of the quadratic term of the polynomial in the arc and {3/n respectively. The condition corres exponent of ponding to (8.2 . 1 9) will therefore be violated for large so that
[g(t)] 11'r� ajn
(-J-)
' l'hcRt' co nditions
a rc
for instance
sntitd1cd
n
if
a ::-::-; c :o.=;
d
and fJ
��
1.
[g(t)] 11n
252
CHARACTERISTIC FUNCTIONS
n
cannot be a characteristic function if is chosen sufficiently large. The function can be used to construct an interesting decomposition. The characteristic function of the distribution
g(t)
F(x) = F(x ; a) * F(lx ; c) * F(lx ; d) f(t) = exp {aeit + ce8it + de4it _ a-c -d}.
is then
This convolution of three Poisson type distributions also admits the factorization = exp [,B( 2 - 1 We conclude from theorem that must have an indecomposable factor and see that a convolution of three Poisson type distributions can have indecomposable factors. Since every factor of is also a factor of we conclude from theorem and from a result of P. Levy ( t) that there exist indecomposable characteristic functions of the form P. Levy (1 937b) considered polynomials of the form
f(t) g(t) e it )] . 6.2.2 g(t) 8.2 .4
f(t)
g(t)
P1 (x) = ax+ f3x 2 -yx3 + bx 4 + cxs P2 (x) = a'x -{3x 2 +yx3 +b 'x4
and
(8.2.20).
and showed that it is possible to determine the coefficients in such a way that = exp and = exp are both indecomposable characteristic functions. Therefore = is the characteristic function of a convolution of three Poisson type distributions and provides an example of the factorization of an infinitely divisible characteristic function into two indecomposable factors. We conclude this section by listing several theorems which indicate that certain functions can be characteristic functions , provided a parameter is suitably chosen. These theorems are somewhat similar to theorem since they can also be used to prove the existence of infinitely divisible characteristic functions having an indecomposable factor.
{P1 (eit) -P1 ( 1 )} {P2 (eit) - P2 ( 1)}
/1 (t) /2 (t)
f(t)
/1 (t) /2 (t)
6.2. 3
Theorenz 8.2. 7. Let = pI be a rational number and suppose that the integers p and are relatively prime and that 1 < p < For given positive numbers A1, A2 andy it is possible to select a sufficiently small positive number that /1 {t) = exp {-yt it A1 (e - 1 ) + A2 (ecxit - 1 ) (eitfq - 1 )} a characteristic function. q
so is
ex
q
2
q.
-v
+
(t) P. Levy (1937b) has shown that a
function
v
of the form exp [P (x) - P ( l )] (P (x) a polynomial) cannot be a generating function un less a term with negative coefiicient is p receded by one term and followed by at least two terms with positive coefficien ts .
253 Theorem 8.2.8. Let oc be an irrational number, 0 < ex < 1 . For given positive numbers A1 , A2 and y it is possible to select sufficiently small positive numbers v and so that the function 2 f2 (t) = exp { -yt +A1 (eit _ 1 )+A2 (ecxit _ 1)-v(errit _ 1)} is a characteristic function. Theorem 8.2.9. Let G(u) be afunction which is continuous and non-decreasing in the interval [b1 , b2] and suppose that G(b2)-G(b1) > 0 and let y be a positive constant. Then it is possible to select sufficiently small positive numbers v and so that the function f3 (t) = exp {-yt 2 + f bt, (ei1" - l)dG(u)-v(eitYJ _ 1)} b is a characteristic function. Theorems 8. 2 .7, 8.2. 8 and 8.2.9 are due to Yu. V. Linnik ; for their proof we refer the reader to Chapter 8 of Linnik ( 1964). Remark. We see from the Remark 1 following theorem 5.5.1 that the characteristic functions /1 ( t), /2 ( t) and /3 ( t) are not infinitely divisible and therefore have indecomposable 'factors. FACTORIZATION OF ANALYTIC CHARACTERISTIC FUNCTIONS
1J
1J
We mention here another open problem of the arithmetic of distribution functions. It is known that many infinitely divisible characteristic functions have indecomposable factors ; however, it is only possible to determine these factors in a few cases. It would be interesting to study methods which would permit the determination of indecomposable factors of infinitely divisible characteristic functions. 8.3
Determination of certain entire characteristic functions by properties of their factors In studying factorizations we disregard the trivial degenerate factors. It is therefore convenient to introduce the following terminology. We say that two characteristic functions are equivalent and write and
/1 (t) /2 (t) (t) (t) /1 !2 r--.1 if /1 ( t) == eiat f2 ( t) where a is a real number. Similarly we say that the second characteristics � 1 ( t) == lo g /1 ( t ) and � 2 ( t) == log /2 ( t) are equivalent (in symbols cfo 1 (t) r--.� � 2 (t)) if �1 (t) == ait + �2 (t). With this notation we can express the fact that two characteristic func tions /1 ( t) and /2 ( t) belong to distributions of the same type (t) by stating that there exists a constant a > 0 such that /1 (t) r--.� /2 (t/a). we show that certain entire characteristic functions can be I n this section
("f")
For the sake of b.-evi ty we will
any
thnt }'1 (t) nnd J� (t)
nrc
of the
smnc
typ e .
254
CIIARACTERISTIC FUNCTIONS
characterized by properties of their factors. We derive first a theorem which is the converse of Cramer ' s theorem.
Theorem 8.3.1. Let f(t) be a decomposable characte1�istic function and suppose that all factors off(t) are of the type off(t). Then f(t) is the charac teristic function of a normal distribution. We prove first that f ( t) is infinitely divisible. Let f(t) = fl (t) f2 (t) be a decomposition of f(t). It follows then from the assumptions of the theorem that there exi$t two positive constants c1 and c2 such that (8.3.1) f(t) � f(cl t)f(c2 t). We apply the same decomposition to each factor on the right-hand side of (8.3.1) and see that f(t) � f(ci t)f(cl C2 t)f(cl c2 t)f(c� t) so that [f( c1 c2 t) ] 2 is a factor of f(t). According to the assumption of the theorem there exists then a positive constant c3 such that [f(c1 c2 t)] 2 l s, then the three numbers a, a - s and a + s have the same sign. We introduce the function if x < a - s 0 Hs (x) = � [O(x) - O( a - s)] if a - s � x < a + s � [0( a + s) - 0( a - s)] if a+s � x. The functions H8 (x) and O (x)- H8 (x) are then both bounded and' non decreasing functions ; therefore ( oo 1 -J-2x2 dH (x) h (t) l 0 such that h, (t) --- !(;) oo (eitx _l - itx 2) 1 +2x 2 dH (x) J - oo 1 +x x oo (eitx!c _ 1 - itxjc2) 1 +2x 2 dO(x). 1 +x x J
255
is a factor of f(t) ; hence there exists a or
e
�
- oo
By a simple transformation of the integral on the right-hand side of this relation we see that
oo (eitx _ 1 - itx 2) 1 + 2x 2 dH, (x) J - oo 1+x . x J (e"'tv _ 1 - 1 it+yy 2) 1 y+y2 2 c12 (1+ c+y2 y 22) dO(cy). It follows that 2 2 x 1 c y + H, (x) C + J 2 ( 1 +y 2) dO(cy) �
ao
- ao
=
C +s
- oo C
He (x) O(x) c(a+s) . [c(a-s), ] O(x), c s) � � c(a+ s) �c� (8 .3 .3) a- s � c � + We see from (8.3 .3) that c tends to 1 as s � 0 ; at the same time the interval which contains all th e points of increase of O(x) shrinks to the point x a ; hence x a is the only point of increase of O(x), and O (x) has the form O(x) As(x - a) (A > 0) . Then 2 1 a + log f (t) A 2 (e"'ta - 1) a and it is easy to show thatf(t) cannot have a proper decomposition (8.3.1 ). This shows that a 0 leads to a contradiction with the assumptions of our theorem, so that a 0 is necessarily the only point of increase of O(x). This
where is a constant. The function increases only in the interval [a - s, a ] , therefore Since a is a grows only in point of increase of it must lie in this interval, i.e., (a a so that a a if a > 0 a+e a-e a a if a < 0. a e =
=
=
•
�
=I=
=
means thatf(t) is the characteristic function of a normal distribution , so that theorem 8. 3 . 1 is proved.
256
CHARACTERISTIC FUNCTIONS
8 2 1)
It follows from Cramer ' s theorem (theorem . . that a normal charac teristic function has only factors of its own type. Theorem is therefore the converse of Cramer ' s theorem and we obtain immediately the following characterization of the normal distribution.
8.3.1
Corollary to theorem 8.3.1. The decomposable characteristicfunctionf(t) is the characteristic function of a normal distribution if, and only if, allfactors of f(t) are of the type off(t). Our next theorem gives a common property of normal distributions , Poisson type distributions and their conjugates.
Theorem 8.3.2. Suppose that the characteristic function f(t) has an infinite set of non-equivalent factors and assume that f(t) has the following property : if f1 (t) and f2 (t) are any two factors of f(t), then either f1 (t) is a factor of f2 (t) or f2 (t) is a factor of f1 (t). Then f(t) is the characteristic function of either the normal distribution or of a Poisson-type distribution or of the conjugate to a Poisson-type distribution. For the proof of theorem 8.3 .2 we need the following lemma. Lemma 8.3.1. If a characteristic function f(t) is divisible by an arbitrary integer power of a characteristic function g(t), then g(t) belongs necessarily to a degenerate distribution. If the conditions of the lemma are satisfied, then f(t) = [g(t)]n hn (t) (n = 1 , 2, . . .) where hn ( t) is some characteristic function. Therefore (8.3.4) I f(t) I == I g(t) I n I hn (t) I � I g(t) In (n = 2, . . .) . We now show that the assumption that g(t) is non-degenerate leads to a contradiction. It follows from the corollary to lemma 6.1.1 that there exists a � > 0 such that I g(t) I < 1 for 0 < t < �- We choose such a t and let n tend to infinity in (8.3.4) and see that I f(t) I can be made arbitrarily small, provided 0 < t < �- This contradicts the fact that f(t), as a characteristic function, is continuous at t = 0 andf( O ) == 1, so that the lemma is proved. We proceed to the proof of theorem 8.3 .2 and show first that the characteristic function f(t) has no indecomposable factors. We give an indirect proof and assume therefore tentatively that f1 (t) is an indecom posable factor of f( t). According to the assumptions of the theorem, every other factor g(t) of f(t) is divisible by some power of f1 (t). We see then from lemma 8.3 . 1 that there exists a highest power of f1 (t) which is a factor of g(t). Let n be the exponent of this power, so that g(t) = [f1 (t)]� h(t). The factor h( t) is not divisible by f1 ( t) it follows from the assumptions of the theorem that h(t) must be a factor of f1 (t), but since f1 (t) is indecom posable, h(t) is necessarily degenerate, so that (8.3.5) g(t) "' [fl (t)] n . t,
;
257
FACTORIZATION OF ANALYTIC CHARACTERISTIC FUNCTIONS
If the characteristic function is non-degenerate it can, according to lemma On the other not be divisible by arbitrarily large powers of /1 hand we see from that every factor of is equivalent to some power of /1 so that can have only a finite number of non-equivalent factors. This contradicts the assumption of the theorem, so we must con clude that has no indecomposable factors. According to theorem is then infinitely divisible. Therefore we can write in the canonical form i X ex p � +
8.3 .1,
(8.3.5) ( t), f(t) f(t)
f(t) (8.3.6) f(t)
.N
(t).
f(t)
f(t) [J (e'-tro 1 - tx 1 + 2 dO(x) . _ 1 x 2) x 2 J OO
"
6.2.2
•
oo
O(x)
We show next-again by means of an indirect proof-that has only a single point of increase. Let us therefore assume tentatively that this is not true and that has the points and as points of increase. We select > so that and construct two functions if X ro
O(x) a 1 a2 8 0 a1 + 8 < a2 - 8 < a; - 8 a3 - 8 � x < a1 + 8 (8.3.7) H1(x) O(x)-O(a1-8) 0 ( a1 + ) - 0 (a1 -8) if a1 + 8 � x for j 1, 2. The functions 0(x)- H; (x) (j 1, 2) are non-decreasing and bounded, so that the functions 2 1 x x u.. fi (t) exp [ J'X) (e - 1 - 1 : x 2) :2 dH; (x)] (j 1, 2) are characteristic functions. Moreover /1 ( t) as well as /2 ( t) are factors of f(t). We conclude then from the conditions of the theorem that one of these factors must divide the other. If /2 (t) would be a factor of/1 (t), then 2 itx 1 /1 ( t) + r 00 ( 8.3.8) /2 (t) = exp { J _ oo (e" " - 1 - 1 + x 2) x 2 d[H1(x) -H2 (x)]} would be a factor of f ( t) and therefore an infinitely divisible characteristic function. However we see from (8.3. 7 ) that H1 (x)-H2 (x) is not mono tone, so that the expression (8.3.8) cannot represent an infinitely divisible characteristic function. Therefore /2 (t) cannot be a factor of /1 (t). In the same way we can rule out the possibility that /1 ( t) is a factor of /2 ( t) and therefore obtain a contradiction with the assumptions of the theorem. This contradiction shows that O(x) has exactly one point of increase. Let x a be this point. If a 0, we see from (8 . 3 . 6) that f(t) is the characteristic function of a normal distribution ; if a > 0 then f(t) is the characteristic function of a Poisson-type distribution ; if a < 0 then f(t) is the conjugate ·
if
e
=
=
=
=
=
00
·t
=
X
==
of a Poisson-type characteristic function. This completes the proof of theorem 8.3 .2. Its converse is trivial, so that it can provide a characterization of the family of all distributions which belong to the type of the normal, the Poisson or the conjugate Poisson distribution .
25 8
CHARACTERISTIC FUNCTIONS
We finally mention a result due to I. A. Ibragimov (1 956b) which gives a characterization of the normal distributions. He considered the class ff of infinitely divisible distribution functions which have the following property : if E ff and if the convolution = Q is infinitely divisible then H is infinitely divisible. Ibragimov showed that the class ff coincides with the family of all normal distributions.
F(x)
F(x)
F*H
8.4
Infinitely divisible analytic characteristic functions In this section we discuss analytic characteristic functions which are infinitely divisible. We have seen earlier that an infinitely divisible charac teristic function does not vanish for real values of its argument and we now extend this remark.
Theorem 8.4.1. Let f(z) be an analytic characteristic function and suppose that it is infinitely divisible. Then f(z) has no zeros in the interior of its strip of regularity . Since f(t) is an infinitely divisible characteristic function, [f(t)] 1 1n is a characteristic function for any positive integer n and is also a factor of According to theorem 8 . 1 . 1 the function [f(z)] 11n is an analytic f(t). characteristic function which is regular at least in the strip of regularity of f(z). If1 nf(z) should have a zero at some point inside this strip, then f(z)] 1 would have a singularity at the point for sufficiently large n, [which is impossible. The statement of theorem 8. 4 .1 cannot be improved. This is shown by z0
z0
·
constructing an analytic characteristic function of an infinitely divisible distribution which has zeros on the boundary of its strip of regularity. Let > > be two real numbers and put == It is easy to show that
w a+ib.
a 0, b 0
w l it/w (1 it/ ) )( f (t) (1 - itja) 2 is an analytic characteristic function which is regular in the half-plane Im (z) > - a and \vhich has tvvo zeros - iw and - iw on the boundary of this region. Moreover it admits the representation Iogj(t) = mit+ I � ( euu _ l - �: 2 ) dN(u) 1 =
where
and
N (u)
=
-2
J� e- at (1 - cos bt)t - 1 dt.
FACTORIZATION OF ANALYTIC CHARACTERISTIC FUNCTIONS
259
2 f(t) is Corollary 1 to theorem 8.4.1. An infinitely divisible entire characteristic function has no zeros. P. Levy (1 93 8a) raised the question whether an entire characteristic function without zeros is infinitely divisible and solved it [P. Levy ( 1 93 7 c)] by constructing an example of an entire characteristic function without zeros which is n·ot infinitely divisible. The characteristic function (8.2.20) is such an example. Moreover our argument in Section 8.2 indicates that it is possible to determine the coefficients in such a manner that (8.2. 20) According to P. Levy's representation theorem (theorem 5 . 5 . ) infinitely divisible and therefore provides the desired example.
represents an entire and indecomposable characteristic function without zeros.
Corollary 2 to theorem 8.4. 1 . The characteristic function of a finite distri bution cannot be infinitely divisible. The corollary follows immediately from theorem 7 .2.3 and corollary 1 to theorem 8.4. 1 . Corollar 3 to theorem 8.4.1. The characteristic function f(t) of a finite y distribution is always the product of a finite or denumerable number of in decomposable factors. From corollary 2 and from theorem 6.2.2 we conclude that f(t) must
have indecomposable factors. It is also easily seen that it can have no infinitely divisible factors since all its non-degenerate factors are entire functions of order and cannot therefore be infinitely divisible. The preceding theorem and its corollaries can be regarded as necessary conditions which an analytic characteristic function must satisfy in order to be infinitely divisible. For instance it follows from theorem that the characteristic function determined by formula ( 5.5 is not infinitely divisible. We used this fact-without proving it-in an example discussed in Section 5 .5. Vv"e now give anoth er application of the theorems discussed in the present section. Let be an infinitely divisible analytic characteristic function which {3. Then has no factor of has the strip of regularity - < lm the form g(t) for which log p - log f3 . - (X < I:
1
.12)
f(z)
=
oc peit� +(1 -p)eitTJ
8.4.1
(z) < f(t) (1 -p)
(z) are obtained from (9.2.14), (9.2.1 3) and (9.2. 1 1 ). We define g�.r > (z) = gq,r (z) -g�� > (z) and obtain easily the second estimate of the lemma. We saw that the functions g�� > (z) and g�.r > (z) are entire functions which are real for real z. The Maclaurin expansions of these functions therefore have real coefficients, so that g��) (x + iy ) and g�� > (x - iy) [respectively g�.r ) (x + iy) and g�.r ) (x - iy)] are complex conjugate for x and y real. Writing
u�� > (x , y) = Re [g�� > (x + iy) ] , u�.r > (x, y) = Re [g�.r > (x + iy)] we have u��> (x, y) = �[ g�� > (x + iy) + g��> (x - iy)] (9.2. 1 5) . ( ) ( . + ) 1 ) ) ( + ( ( ) ( x y x, [ z y - 2 gq,r gq,r - zy)] . Uq,r The functions on the right-hand side of (9 .2. 15) are entire functions. One can therefore consider the equations ( 9.2. 1 5) as definitions of u�� > (x, y) and u�.r > (x , y) for complex x (and fixed y). We use the estimates of lemma 9 .2.2 and see that 0 { 1 x 1 5 exp [N (Re (x)) 2] } if Re (x) > + < ) (9 .2. 1 6a) uq,r > (x ' Y = O( J if Re (x) � if Re (x) � 0( 1 < ) ( (9.2. 1 6b) y uq,r > ' - O {J x exp [N (Re (x)) 2] } if Re (x) We introduce the functions J(q,t (x) = u�1) (x, - u�i > (x, 2nttq.l ) Kq,2 (x) = u�.2 > (x, u�.2! (x, 2n�-tq.l ). Clearly these are entire functions , and we see from (9 .2. 1 6a) and (9.2. 1 6b) that they admit the estimates O { I x l5 exp [N (Re (x) ) 2] } if Re (x) > K (9.2.1 7a) q,l (x) = if Re (x) 0( if Re (x) � (9.2. 1 7b) Ka.2 (x) = O { J x 1 5 exp [N (Re (x)) 2] } if Re (x) as I x I -+ oo. For the study of the functions Ko..r (x) ( r = 1 , 2) we need two analytical results. The first of these can be derived from lemma 8.2. 1 .
{
X
X
{ X 1 5) { X 1 5) 15
{ 1 X 1 5) {0( 1 X 1 5)
0 0 0 < 0.
0) 0)-
0 �0 0 0 and where the series converges1 uniformly on every bounded set. Then a1 = b1 = 0 for j > w [k T (2n) -· ] . The proofs of lemmas 9.2.3 and 9.2.4 are given in Sections E3 and E4 �
z
�
0
�
�
;
00
oo
=
respectively of Appendix E. We return to the investigation of the functions Kq.r (x) and prove the following statement.
Lemma 9 .2. 5 . The functions Kq,r (x) (r 1 , 2) are polynomials of degree not exceeding 5 . Since uq, r (x, y ) = u��> (x, y ) + u�.r > (x, y) we see that Kq,1 ( x) = [uq,1 ( x, O ) - uq,1 (x, 2nttq, 11 )] - [u�. 1 > (x, 0) - u�. 1 > (x, 2nttq.l )] . =
In view of (9.2.7a) and (9.2. 1 6b) we obtain for real x the estimate as x tends to + oo , (9 .2. 1 8) We see from (9.2.1 7a) that Kq.1 (x) = O {l x l 5 exp [N (Re (x)) 2] } ( l x l -+ ) in the half-plane Re (x) � The conditions of lemma 9.2.3 (with / = Kq . l ; = = 5, = = N) are satisfied, so that (9.2. 1 8) holds in the half plane Re (x) � We see from (9.2. 17a) that (9.2. 1 8) holds also for Re (x) � so that (9.2. 1 8) is valid in the entire x-plane. Let
a c
0,
b 0, d 0.
oo
0.
Kq, l (x) = � 00
i=O
ax i
i
be the Maclaurin series for Kq,t (x) and consider the function
H(zv)
=
K(J.I
(e'0)
=
2: 00
.1 ; ()
a1 ewi .
272
CHARACTERI STIC FUNCTIONS
Since {9 .2. 18 ) holds for all real or complex x, we conclude that = O {exp [5 Re (w)] } . The function H(w) therefore satisfies the conditions of lemma 9.2.4 (with = 2n, b; = = k = 5 , w = 5) so that i = for j > 5 . Therefore Kq. t (x) is a polynomial of degree not exceeding 5 . The statement con cerning Kq,2 (x) is proved in the same way. = 1 , 2) given by lemma The information concerning the Kq,r (x) 9.2.5 permits us to get more precise results on the functions g�}> (x) and g�. 2 > (x) .
H(w) 0,
f H, T
a 0
(r
Lemma 9.2.6. The functions g�}> (x) and g�.2 > (x) admit the expansions gq< +,l ) (x)
(9 .2. 19a)
=
g�. 2 > (x)
( 9.2. 1 9b)
� (a.1�q. 1> + bJ�q. l> x) exp [u. rvq, l J'x] + Sq,l (x) j= l 00
00
� (aj:� + bJ;� x) exp [flq. 2 jx] + Sq, 2 (x) j= l bj:� (9.2. 19c) I a�� I + I b]:; = O {exp ( - kj 2) } ( j --+ oo ; = 1 , 2) som e k > 0. Sq,r (x) e =
where the real constants a�:� and satisfy the condition I r The ar polynomials of degree not exceeding 7 and for have real coefficients. We p rove only the statement concerning g�} > (x), since the statement concerning the second function is proved in the same way. To simplify the notation we write in the proof of formula (9.2. 1 9a) .A. and S(x) instead of g�}> (x), 2ni/flq. t and Sq, t (x) respectively. After the completion of the proof we revert to the original notation. It follows from the definition of tl1e function Kq. t (x) and from (9.2. 15 ) that Kq, t (x) = �}> (x) - � {g�} > (x + 2n flq. 11 ) + g�} > ·(x - 2niflq. l ) } . Since Kq. t (x) is a polynomial of at most fifth degree, we obtain (using our simplified notation) the relation
h(x),
g
i
h(x + A.) - 2h(x) + h (x - A.) � ci xi where the c i are constants (which depend on the suppressed subscript q). We choose constants c; (also depending on q) such that the polynomial P (x) � c; x i satisfies the equation j=2 P (x + A) - 2P (x) + P (x - A.) � c i xi . =
=
j =O
7
=
The function h1 (x) = (9 .2.2 )
0
5
h(x) -P(x)
5
j=O
CHARACTERISTIC FUNCTIONS WITHOUT INDECOMPOSABLE FACTORS
then satisfies the equation
273
h1 (x+A)-2h1 (x) + h1 (x - A) 0 so that the function h2 (x) h1 (x)-h1(x-A) is periodic with period A. From the definitions of the functions h 1 (x) and h2 (x) and from lemma 9.2.2 we see that h2 (x) is an entire function and that for large I x I x F exp [N ( Re (x)) 2] } if Re (x) > 0 { {0 1 h2 (x) 0( I x 1 7) if Re (x) 0. The conditions of lemma 9.1. 2B are satisfied, so that (9.2.2 1) h2 (x) � d1 exp (flq.1jx) where d1 [ exp ( - kj 2 )] for some k. (We write here again d1 instead of 0 d.) function h (x)-A - 1 xh (x) is periodic with period It is again The A. 1 2 possible to apply le a 9 . 1 .2B and we get (9.2.22) h1 (x)-A - 1 xh2 (x) � d1 exp (P,q,1jx) where d1 djq> O [exp ( - kj 2)] for some k. We now return to the original notation and see from (9 .2.20), ( 9.2.21 ) and (9. 2 . 22) that g (x) � (a�Jq.l> + b�.1q. 1> x) exp (rq,lJ·x) + � c'.1. xi where O[exp ( - kj 2)] I aj;i I + I b]:l l for some k 0. The statement of the lemma follows from the fact (estab lished in lemma 9.2. 2) that g�� > ( ) is real for real x. Let tq (z) g(z) - g�}> (z) exp (flq.t z) -g�.2 > (z) exp (flq.2 z) . Using relations ( 9.2.9 ) and (9. 2 .10), we obtain the following two repre sentations for tq (z): exp (flq, 1 z)- g�. 2 > (z) exp (ttq. 2 z) . g (z) (z) > � tq . 1 { (9·2· 23) tq (z) g�1 l (z) exp (,u0,2 z) -g�.1 > (z) exp (flq.l z) . =
=
x
=
=
=
We apply the estimates of lemma 9.2.2 to these representations and see that � Re 1 5 exp if Re ta ( ) = � Re ( )] } if Re z 1 5 exp t oo .
z 0{ 1 z z 0 {1
Cuq.l (z)] } [pq,2 z
(z) 0 (z) 0
274
CHARACTERISTIC FUNCTIONS
We introduce the expansions (9 .2. 1 9a) and (9 .2. 19b) into (9 .2.23) and see that
= r�=l j=� 2 (cj;� + zd}:�) exp (flq,,.jz) + Lq (z) (9 .2.24) where Lq (z) = tq (z) + Sq,t (z) exp (flq.t z) + Sq,2 (z) exp {flq.2 z) where cJ,qr\ = a�Jq-> l .r and dJq.r> = b.:;�q-) l ,r . As a consequence of (9.2. 1 9c) we have the estimate (9 .2.24a) I cj;� I + I d]:; I = O[exp ( - kj2)] (j --+ oo) . 2
g ( z)
00
�
�
We also note that Lq (z) is an entire function which is real for real z. Using the estimates for tq (z) we see that z j 7 exp [#q,t Re (z) ] } fo r Re (z) . (9 2. 24b) Lq (z) z 17 exp [#q,2 Re (z)] } for Re (z) � as z --+ We introduce the functions hq,t (z) g�} > (z) - �o..9q,t (z) exp (flq.t z) (9 .2.25) hq,2 (z) g�.2 ) (z) - sq,2 (z) exp (flq.2 z) . It follows from (9.2. 19a) and (9.2. 1 9b) that
{ 0 { = 0{ 11
I I oo.
{
(9 .2.26)
;::: 0 0
= =
hq,r (z)
so that
= j= 2 { cj:� + dj:; z) exp (flq,,.jz) � 00
2 (9.2.27 ) � hq.r (z) + Lq ( z) g( z) r=l where Lq (z) is the function defined in (9 .2.24). The functions hq,r (z) are entire functions and are real for real z . Let z t + iy ( t, y real) and write (9.2.28) Hq,r ( t, y ) hq,,. ( t ) - ! [hq,r ( t + iy) + h q,r ( t - iy) ] .
=
=
=
The right-hand side of this equation is for fixed y an entire function of t. The function Hq,,. ( t, y) can therefore be continued into the complex plane, and we write Hq,r ( y) for its analytical continuation. We consider the function Hq,r y) for fixed real y and complex and use the estimates of lemma 9.2.2 and formulae (9.2 . 25 ) and (9.2.28) and see that for --+ oo , > 0 1 7 exp [N (Re ( )) 2] } for Re (9.2.29a) Hq,t � for Re 17) for Re (9 .2.29b) Hq,2 ( 17 exp [N (Re (x)) 2] } for Re ( x)
q + { 9 .2.32a)
Hq, t { t, 2ntts. l)
and
=
s
1
0[exp {tts- 1.1 t)]
(
s
=
q + 2, q + 3 . . . )
Hq. 1 ( t , 2npq_;1,1 ) = O [t 7 exp (flq. 1 t )] as t -+ + We see from (9.2.29a), (9.2.32a) and (9.2.32b) that the q + 1 , the conditions of functions Hq.1 ( t, 2ntts. l ) satisfy, for integer lemma 9.2. 3 and we conclude that
(9.2. 32b)
(9.2.33a)
oo.
s�
Hq, l (x , 2nps.l )
=
0 { 1 x 1 7 exp [tts- 1.1 Re (x)] }
[ Re ( x) � 0, and we see from (9 .2.29a) that, for I x I -+ oo ,
0( 1 1 7)
s
=
q + 1 , q + 2, . . .]
for Re (x) � 0. Hq, 1 ( x , 2ntts. l ) = x By means of a simple computation we obtain from (9.2.28) and (9.2.26) the representation
(9.2.33b)
(9.2.34)
Hq. 1 (x , y )
=
2}�2 (cj�l + dj';l x) (sin /lq,tyr exp (flq.dx)
+
'., yd(a) st• n (l"a .11Y• ) exp (/lq, X)
£.J :1 � 2 00
j)
t
)
•
•
276
CHARACTERISTIC FUNCTION S
We substitute y = 2n fls. 1 (with s == q + q + 2, . . . ) into . 2 .34 ) and obtain a series of the form treated in lemma 9.2.4 with coefficients a; =
9. 5
( 2.3 )
T
b;
/ 0 0
=
/
1,
( (
2) + dj�l 2n sin 2/la.dn fls. l fl s.1 )2
(9
2cJ�l sin flo..dn fls.1 2dJ�l sin fla.dn fls.1
and = 2n flq. r Then w = fls - 1 • 1 flq.l and the coefficients aj and b; vanish if j > fls - 1 • 1 flq.l (s = q + q + 2 , ) It follows from ( 2.3 ) th at • -l for fls- 1 .1 flq-.11 d(j.q1) ) fls .1 flq.1 (s = q + ' q + 2 ' and therefore also q • -1 for #s - 1 , 1 flq-.11 fls .1 flq.1 (s = q + ' q + 2 ' . . . ) . cj( , 1) = The coefficients cj�i may be different from zero only if j belongs to the set ] 1 = {flq +p .1/flq. 1 };' 1• (It follows from the assumptions of the theorem that J1 is a set of integers.) We show next that dj�i = even for j E J1 • We carry an indirect proof and assume that d(:f ¥= We put y = 2njflj.l in .2.34) and see that
1,
9. 5 1 ... 1
.... <J < <J
& ., .c0r 1 = � c� 2 3 7) 1' ""' J .1 Am- 1 , 1 • • • flq.1 is valid. We give an indirect proof for and assume that for some does not hold. Let = flm - 1 • 1 flq.l integer m � q + 2 the inequality be the corresponding subscript. Then Zn sin flm 1 ' 1 n (1 + o ( l )) exp (Jlm - 1 , 1 t) as t _,. co . = 2c�i Ha.1 t , flm .1 flm.1 It follows then from .2.3 1 ) , .2.3 ) and . 2 .2 b ) that n sm flm - 1.1n ( l + o ( l )) exp ( - l .l t) = 2c� � u( t , O ) - u t, Z flm i flm.1 flm.l as t + oo . The last relation leads to a contradiction with ( .2 . ) so that the validity of .2.3 7) is established.
(9
0
(
0
-+ (9
. ..
�
)
(9.2.37) (9.2 .37)
y
(
(
i
(9
)
(9 0
(
p
(9 9
y
9 5,
277
CHARACTERISTIC FUNCTIONS WITHOUT INDECOMPOSABLE FACTORS
The function Hq. 2 (x, y) can be treated in a similar manner. One obtains an expansion corresponding to (9.2.34) with coefficients cj�� and One can again show that j;� = 0 for j = 2, 3 . . . while the coefficients cj;i may be different from zero only if j belongs to the set ]2 = {ttq + P. 2/flq. 2}; 1 and where 2 (9.2.37a) 0 � cj;i � Am - 1 .2 for j = Pm 1 flq.2 for m = q + 2, q + 3 . . . . (q> for p = rvm,r rvq,r We write cPeq.r> = Am,r and m = q + 1 , q + 2, . . . and obtain from (9 .2.26) and (9.2.27) the representation
dj;J .
d
-
·
u. ju.
(9.2.3 8)
00
2
g (z) = � � A��r exp (flm.r z) + La (z) r=1 m=q + 1
where (9.2.3 8a) 0 � A.��r � Am.r (m = q + q + 2 , . . . ) . The representation is valid for each q = 0, + 1 , + 2, . . . . We select two arbitrary integers q1 and q2 , q1 < q2 and write (9.2.3 8) for q == q1 and q = q2 and subtracting the two equations we obtain (9.2.39 ) 00 00 (q > q1) - Am.2 (q s> l exp rvm , l z) = - � [A(m.2 s ] exp rvm,2 z) � [A.(mq,1>1 - Am.2.J m =�+ 1 m =�+ 1 2 q2 - � � ).��� exp (ttm.r z) - Lq1 (z) + Lq (z) . r=1 m=qt+1 We write A(z) for the sum on the left of (9.2. 39) and see from (9.2.38a) and from condition (9.2. 1 ) of theorem 9.2. 1 that for Re (z) � 0. A (z) = To estimate A(z) for Re (z) 0 we use again (9.2.3 8a), (9.2. 1 ) , (9.2.24b) and the fact that #m.2 < 0 and obtain for the four terms on the right of (9.2.39 ) the estimates z exp [flq1, 1 Re (z)] } 0(1 ) , O {exp [flq2,1 Re (z)] and z j7 exp [flq2,1 Re (z) ] } respectively. Therefore z exp [flq2, 1 Re (z)] } for Re (z) � 0. A(z) == We see that the function A(z) satisfies the conditions of lemma 9.2.4 with k = fla2.u a1 == 0 if j is not one of the integers {ttm. 1/ftq2,1 }:- qa + 1 , h i = 0 for j = 0, + 1 , + 2 . . . = 2n /flq2, 1 . Then w = 1 and we see that � ( q1) � ( q2 ) .cOI'" m - q 2 + 1 ' q2 + 2 ' • • • • 1 ' ll.m,l - ll.rn, l = 0 Similarly one can prove that � ((/ .) _ 1Lw, � (q.2) - 0 f0 1 '"'l = �(f2 1 1 ' q2 + 2 ll.m.2
1,
(u.
(u.
2
0(1) �
0{1
}, 0{ 1 1 7
0{ 1 1 7 ;T
-
..
-
..
m '
� .-
'
•
•
•
·
278
CHARACTERISTIC FUNCTIONS
Since q1 and q2 are arbitrary we conclude that the A��r do not depend on q ; we therefore write Am.r instead of A��r and obtain the following representation for g (z) : (9.2.40)
00
00
g(z) = � � A.m .r exp Cum .r z) + Lq (z) r=l m =q + 1
where (9.2.40a) 0 � Am .r � Am.r ( m = 0 , + 1 , + 2 , . . . ) and where this representation holds for all integers q. We introduce the function (9.2.4 1 )
(fi (z) =
� r=1
�
J:m .r
(eZ!Jm,r - 1 1Zftm� ) . + flm .r _
m = � oo We repeat the reasoning used in the proof of theorem 9 . 1 . 1 and see that �(z) is an entire function and that �(z) = 0 {1 z l 2 exp [N (Re (z)) 2] } (9.2.42) where N > 0 is a constant. We introduce the function L(z) = g(z) - �(z) . To prove theorem 9.2. 1 we must show that L(z) = jiz 2 + /Jz where p is real while ji � 0. The first step in the proof is the demonstration that L(z) is a polynomial of degree not exceeding We see from (9.2.42) and lemma 9 .2. 1 that there exists a positive con stant A such that I L(z) I � A l z 1 3 for Re (z) = 0. According to formulae (9.2.40) and (9 .2.40a) we can represent L (z), for each q = 0, + 1 , + 2, . . . , as
3.
L(z) =
J:m,r ( 1 + flm .r : ) ± � r=1 m=q+ 1 1 + flm .r £ J:,.,, ( - 1 - flm.� ) + Lq (z) ± 1 + flm .r r=1 m = oo
e/-lm.rz
= � 1 + � 2 + Lq (say) . The sum � 1 is a linear function of z and we have � 1 = 0 ( 1 z l ) as l z l -+ oo . We next use lemma 9 . 1 . 1 and estimate � 2 in the same way in which we estimated s 1 in the proof of theorem 9. 1 .2, and obtain � 2 = O {l z l 2 exp [/,tq, 1 Re (z)] } + O { I z l 2 exp (/,ta. 2 Re ( z)] } ( l z l -+ oo) .
CHARACTERISTIC FUNCTIONS WITHOUT INDECOMPOSABLE FACTORS
279
Using these estimates for � 1 and � 2 and the estimate (9.2.24b ) for Lq (z), we conclude that z exp Luq.t Re (z)] } if Re (z) � 0 ( 9.2.43 ) L (Z) = z I ' exp [#q 2 Re (z)] } if Re (z) � 0, as I z I -+ We apply lemma 9.2.3 to the function L (z) (with M1 = a = 3, = = flq.u c = 7) and see that J L(z) � I z exp [,uq t Re (z)] if Re (z) � 0 . I f one applies lemma 9.2.3 to the function L( - z) (with M1 = a = 3, b = = - flq. 2 , c = 7) one sees that z 1 3 exp Luq.2 Re (z)] if Re (z) � 0 . L (z) � We note that the constant A in these estimates is independent of q and that q can be chosen arbitrarily from the positive or negative integers. We therefore let q tend to we finally and since lim flq.r == 0 ( r =
b d
d
{0{0 { 1 1 7 1
oo.
I A 13
I
.
A,
,
A,
I AI
obtain the estimate
oo,
q-?-
1 , 2)
00
A 13
I L (z) I � I z which is valid for all z, so that the entire function L(z) is necessarily a polynomial of degree not exceeding 3 . We note that L(z) is real for real z and that L ( O) = 0 ; therefore L(z) = Jz3 + jiz2 + /1z, where (J, r and J are real constants. we had u( t , y) = Re [g(t + iy)] , so that u( t , 0) - u (t, y) = g(t) - Re [g( t + iy)] or since g(z) = {1(z) + L(z) u (t, 0 ) - u(t, y) == f(t) - Re [f(t + iy)] + L(t) - Re [L( t + iy)] . It is easily seen that L(t) - Re [L(t + iy)] = (3 Jt + y)y 2 • I f one of the relations J = 0, y � 0 were not satisfied then we could find a t such that 3Jt + y < 0. We fix such a value of t and see from ( 9.2.41 ) that �(t) - Re [f(t + iy)] = o(y 2) as y -+ Hence, for such a fixed t and y -+ oo , we would get u (t, 0) - u(t, y ) = o( y2 ) + (3 Jt + y)y 2 -+ This contradicts ( 9.2.2 ) and we see that necessarily J = 0, y 0. There fore
oo. oo.
1
0 . If the spectrum is denumerable we assume that the conditions (9 . 1 . 3a) and (9 . 1 .3b) are satisfied. Suppose that the positive Poisson spectrum contains at least two points, let and p / > tt be two frequencies of the positive spectrum and let A and A ' be the corres =
1 "' - 1
�
#
ponding energy parameters. We show next that the quotient � = #Itt' is a rational number. The characteristic function f(t) then has a factor = exp /1 If � is irrational then it follows from theorem that /1 and therefore also has an indecomposable factor. This contradiction shows that � is necessarily rational, say � = /q, where and q are integers and can be assumed to be relatively prime, < q . We apply the reasoning used to conclude that /1 and therefore also before and use theorem has an indecomposable factor unless p = 1 . The negative Poisson belongs to the class 2. spectrum is treated in a similar way, so that
{- yt 2 +A(eit�t _ 1)+A' (eitp,/a._ 1)} . 8.2.8 (t) , f(t), p p p 8.2.7 (t), f(t), f(t) Remark. The presence of a Gaussian component is essential since y > 0 is necessary for the validity of theorems 8.2. 7 , 8. 2 .8 and 8.2.9. (t)
9.4
Infinitely divisible characteristic functions with bounded Poisson spectrum In this section we study the factorization of infinitely divisible charac teristic functions with bounded Poisson spectrum. We also derive sufficient ( •) I n order to usc (9. 3 . 1 ) in case of a dcn u rnerable
the poss ibil i ty thnt on l y
n
fi n i te n u r n b e r o f e n ergy pan1n1ctcrs as
well
finite sp ectru m is positive .
as a
we
admit
28 2
CHARACTERISTIC FUNCTIONS
conditions which assure that a characteristic function with bounded spectrum belongs to 0 • We use for the spectra the notations of Section 3 . 7 and have to supple ment these by introducing a convenient notation for the vectorial sum of identical summands. We define the symbol ( A recurrently by writing (1)A A and ( A ( - 1 A ( for 2, 3, . . . . We also write
I
=
)
( oo A
==
n) ( n)A . l n=
n ) + )A
=
U 00
n
n)
=
We need the following lemma :
Lemma 9.4 .1. Let A be a closed set on the real line which is contained in the finite interval [a, b] , where 0 < a < b < oo. Then ( oo)A is a closed set. We note that (n)A [na, nb], and since a > 0 any finite interval can intersect at most a finite number of the sets ( n )A . Let x ( oo)A ; then there c
E
exists a sequence of points {x7c} in ( oo )A which converges to x. The interval (x - 1 , x + 1 ) contains therefore almost all elements of this sequence {xk}· 1 ) intersects only a finite number of the However, the interval (x - 1 , sets (n)A . Therefore there exists at least one set ( A which contains an infinite subsequence of the {xk}· The set is closed (t so that x E ( A and therefore also x E ( oo )A ; hence ( oo)A is closed.
x+
(n)A
n)
)
n)
Theorem 9. 4 . 1 . Let f(t) be an infinitely divisible characteristic function without normal component and which has a Poisson spectrum A such that 0 < a sup x < oo. Then any factor f1 (t) of f(t) has the inf x < b form /1 (t) exp [iyt + J� (eitu - l )dN(u)] . where N (u) is a function of bounded v-ariation which is non-decreasing in the half open interval [a, 2a) and which has a spectrum �o.9. N [( oo)A] [a, b] . The constant y is real. Without loss of generality we can assume that f ( t ) is given by f(t) exp [ J� (eitu - l ) dN0 (u)J . (9.4.1 ) where N 0 (u) is non-decreasing and has the spectrum �o.9. N0 = A . Then f(t) c{ 1 + kil [ J� eitu dN0 (u)T /k!} where c exp [- J � dN 0 (u)J . Let F(x) be the distribution function =
=
a: E A
a: E A
=
c
n
=
=
=
(t) We have seen in Section 3 .7, p . 5 7, that the vectorial sum of two closed an d bounded
sets is closed.
283
CHARACTERISTIC FUNCTIONS WI THOUT INDECOMPOSABLE FACTORS
corresponding to f(t) ; then (9.4.2 )
x c{e(x}+ k� Nf (x)/k !}
F( ) =
(x)
Here N�· denotes the k-fold convolution of N0 with itself. We see from lemma 3 .7 .4 that SN:· == (k) SN0 = (k)A == (k)A . It follows from (9.4. 1 ) that (9.4.3) �9F [ ( oo )A] {0} where {0} is the set containing only the point 0. We assume now that f(t) admits a decomposition f(t) = /1 (t) /2 (t). Then * F2 F (x) = F1 where F1 and F2 are the distribution functions corresponding to /1 {t) and f2 {t) respectively. We see from (9.4. 1 ) and from the assumption concerning the spectrum A of N 0 (u) that log f(t) is an entire function of order 1 and type not exceeding Therefore f(t), and hence also /1 (t), is an entire characteristic function without zeros. Moreover, we conclude from corollary 1 to theorem 8 . 1 .3 that log /1 (t) is an entire function of type not exceeding We see from lemma 3 .7 .4 that -
�-
(x)
(x)
(x)
(x),
b.
b.
(+
(+
) SF 2 ; ) SF2 � �9F1 since [0, oo) � SF we conclude that SF1 and �9F2 are both bounded from the left. It is no restriction ( t) to assume that the infimum of �9F\ is the Point 0 . Then 0 E SF1 and 0 E �9F2 , so that �9F = �9F1
+
(9 .4.4) �9F1 u SF3 c SFl ( ) SF2 c SF . We see from (9.4.3) that the point 0 is an isolated point of SF . Let = ( - a1, 0) u (0, a1) for the union of the two open a1 < a and write intervals ( - a1, 0) and (0, a1) . We see from (9.4.3 ) that does not contain any points of �9F, therefore c �9�. We conclude from (9.4.4) that c �9�1 s o that the point 0 is an isolated point of �9F1 • This means that F1 has a discontinuity at the origin. Let be the saltus of F1 at 0 . Then F1 = is a non-decreasing function of where > 0 and v.rhere bounded variation such that (9 .4.5) c ( oo)A c [a, oo). The characteristic function f1 {t) of F1 (x) is therefore given by
V
ds(x) + G(x), Sa
V
d
/1 (t) = 8
("!")
��
V
d
G(x)
d + f: ei1"' dG(x).
V
(x)
'fhis can be shown by replacing F1 (x) by F1(x �j- 8) and F.,.( x) by F2(x - 8) where I ext [F1] .
284
CHARACTERI STIC FUNCTIONS
We select a positive real number n so large that
G11(x) = J : e -rrvdG(y). Let t be real ; then (9.4.6) J � eit:n dG17 (x) J � dG17 (x) < d
write
J � e-'Yf.ll dG(x) < d and
1 (t)]k1 [4>n(t)]knj(k1 ! . . . kn !). •
•
•
We see therefore that the expression on the right of (9 .4. 1 6) is the co efficient of in the expansion of (9 .4. 17) exp It follows that (9 . 4 . 1 7 ) is, except for a constant positive factor, a characteris tic function, provided > 0. We also see from (9.4. 1 6) that
ym
[y4>1 {t)+y 2 4>2 (t)+ . . . +yn 4>n( t)] . y J eu., dG(x) = cfo1 (t), so that 4>1 (t)/4>1 (0) is a characteristic function. Since, according to our assumption, /1 (t) is a factor of f( t), there exists a characteristic function /2 ( t) such that (9.4. 1 8) /1 (t)/2 (t) = f(t) and we can repeat the earlier reasoning and show that (9.4. 19) /2 ( t) = C exp {i�t + 1p1 (t) + VJ2 (t) + . . . + (t) } . Here C and � are constants while the functions VJm ( t) have the form (t) = Jf eu"' aN (x) (m = 1 , 2, . . . , n), A
1f'n
'lflm
(m) A ,...,
[a,lJl
289
CHARACTERI STIC FUNCTIONS WITHOUT INDECOMPOSABLE FACTORS
where N is a function of bounded variation such that
[a, b] [ ml.J1 (m)Al We see from (9. 4 .18) that log /1 (t) + log /2 (t) = log f(t) or, in view of (9.4.14) and (9. 4 .19), [4>1 (t) + VJ1 (t)] + [4>2 (t) +VJ2 (t)] + . . . + [4>n (t) +VJn (t)] = log /{t) . The spectral function of f(t) has the set A as its spectrum, while 4>m (t) + "Pm (t) has (m)A as its spectrum. The sets (m)A are pairwise disjoint, and we conclude easily that (9.4.20) VJm (t) == 4>m (t) (m == 2, 3 , ) As in the case of (9. 4 .17) we conclude from (9. 4 . 20) that, except for a constant factor , exp { YVJ1 (t) - y 2 1> 2 (t) - . . . - yn 4>n (t) } is a characteristic function, provided y > 0. But then the functions Y4>1 (iv) + Y 2 4>2 (iv) + . . . + ynn 4>n (iv) YVJ1 (iv) -y 2 4>2 (iv) - . . . - y 4>n (iv) are convex functions of the real variable if y > 0 . This is only possible if 4>; ' (iv) == 0 (j == 2, ) Since the entire functions 4>1 (t) are Fourier - Stieltjes transforms of func tions of bounded variation we see easily that the functions 4>1 ( t) (j = 2 , . . . , m) reduce to constants. We put (9.4.2 1) 4> (t) == 4>, (0) == (j == 2, . . . , m) , and substituting (9 . 4 . 2 1) into (9. 4 .14) we see that /1 (t) = C exp [iyt + 4>1 (t) + + . . . + en] or /1 (t) == c1 exp [iyt + 4>1 (t)]. Similarly we obtain from (9. 4 . 2 1 ), (9. 4 . 2 0) and ( 9. 4 .19) /2 (t) == C2 exp [i�t + (t)]. The statement of the theorem follows ftom the fact that the functions (t) and 1p1 (t) are , except for a positive constant factor, characteristic 4>1functions. Remark 1. Theorems 8.2.6 and 8.2.5 are particular cases of theorems 9.4.2 and 9.4.4 respectively. Remarl� 2 . Extensions of theorems 9.4.1, 9.4.2 and 9.4.3 can be o n Cu s (1968) in I. V. Ostrovskii (1966). SN
n
c
. . . , n .
-
v
. . . , n .
c1
1
C2
'lfJ 1
in
ppcn
and
f
u
d
29 0
CHARACTERISTIC FUNCTIONS
9.5 Theorems concerning certain factorizations The methods used in the last three sections make it possible to derive results concerning the possible factors. of certain infinitely divisible distri butions. We list in the following a typical result.
Theorem 9.5 . 1 . Let f(t) be a characteristic function which admits the repre sentation , og f(t) i-1 { :�: (aVT +bP,it) exp (i#,t) + E 1 Am, exp (ivm,, t)} + L(it) where the parameters occurring in this representation and the function L(z) satisfy the following conditions: (a) �1 > 0, � 2 < 0 ; r 1, 2) and (b) n1 and n 2 are integers such that < nr < ( n1�1 > n2 �2 ; nr and n; > 0 (r 1, 2) [if n; nr (c) n� and n� are integers, n; ,nr- 1 then the sum � is omitted]; ( d) the coefficients apr and bpr are real, mr 0, vm1 0, vm2 < 0 ( 1 , . . . ). Moreover , ( d1) v1r n;. �r (r 1, 2) , ( d 2 ) vm + 1 .rf vmr is a natural number greater than 1, O [ex (- kv!1) ] ; (d3) for some k > 0 we have Amr (e) L( z) is an entire function which is real for real z and satisfies the estimate (as I z I ) 2 exp [� � Re (z)] } if Re (z) 1 1 { 0 z 0{ L(z) O { J z J2 exp [�� Re (z)] } if Re (z) 0, where �� and �� are real numbers such that max [n 2 � 2 , (n 1 - 1 )�1] � �� < n 1�1 and n 2 �2 < �� � min [(n2 - 1 )�2 , n1 �1]. Let f1 (t) be a factor off (t). Then f1 (t) has the form log /1 (t) ;f,1 {:�n� (liP,+ bVT it) exp (i#, t) + }}_ .%m, exp (ivmr t)} + L(it ) , 1 where the apr and bpr are real constants and where the coefficients J:mr satisfy the inequality 0 � �r � Amr· The function L(z) is entire, realfor real z, and the estinzates { 1 1 3 exp [�� Re (z) ] } if Re (z) 0 z 0{ L(z) O { J z J 3 exp [�� Re (z)] } if Re (z) 0 hold for I z I I
=
oo
oo
�
m ==
==
A
>
�
==
==
p
-+ oo
�
( t) = log f ( t) be the second characteristic of F (x) we see then from
Fi( x)
oo
-
�
oo
;
(10. 1 . 1 ) that
hence oo
- 00
Kj 2, . . . ad inf. ) .
We
write
ooJ
tx dFj(x) � z
1 (j = 1 , 2, . . . , s) . f 2oc, J for the jth cumulant of and note that K2i - l = 0 (j = 1 , It follows from the preceding inequality that sin 2 21>(t) K2 ) � - 2 = + o ( 1 ) as -+ 0 . 1 2 4 sin 2
-
rfo( t) f(t)
.
§(tx) dFi (x t
t t OCj OCj It follows from Fatou's lemma [see Titchmarsh ( 1 939) , section 10. 8 . 1 ] that the second moments of the distributions Fi (x) (j = 1 , 2, . . . , s) exist. We show next by induction that the distributions Fi(x) have finite moments of all orders. We assume therefore that the distributions Fi (x) (j = 1 , 2, . . . , s) have moments of order 2k and show that this im ·
- oo
-
plies the existence of the moments of order 2k + 2. We differentiate equation (10 . 1 . 1 ) 2k times and obtain on the left side a sum where each term contains derivatives of the We arrange the terms on the left side into three groups and write (10. 1 .3) S1 + S2 + Sa = j(2 k> Here
fi (t).
(t).
(t) (t) (t)
(10 . 1 .4)
sl
JJ2k) .� rx; f
{ nm fa": . fa, fa , where each is one of the integers 1 , 2, . . . , s and where the positive integers 1 , , satisfy the relation and
( t)
=
2 ) �r > [ t t t) /�1 ) ) J 2 [ [ t (J ) (t) J (t) J . • Jam ( t) J
a1 r r2, , rm nh n 2, (10. 1 .5) �j = l n1 r1 = 2k . •
•
•
•
•
•
m
nm
We see easily from (10.1 .3) that Sa S1 (0) S2 ( 1 0 . 1 .6) + +
k> (0 ) S1 (t) -2 (2 t) (t) -2 Sa ( 0) /(2k> (t) -f(2 t t [ [2 It follows from the corollary to theorem 3 . 1 .3 and from the definition of the functions S2 ( t ) and Sa (t) that Sa (t)- Sa (0) S2 (t) d an t2 t2 tend to finite limits as t goes to zero. Moreover, we conclude from the fact that f ( t ) is an analytic characteristic function that f2k) ( t)- f(2k) (0) liiil t2 exists and is finite. Hence this is also true for 1. sl (t)- sl (O ) Im i2 . We see from (10.1 .4) that sl (t) - sl (o ) � . /J 27c> (t)- fJ 2k> (O) � . +�2lc> (t) !1 (t)- J( t) = t2 t 2f:i\t) j=l j= l t2 It is easy to see that the second sum on the right of this equation tends to a finite limit as t goes to zero ; therefore � !J2k> ( t) -jj2k> (0) ) 2 � f oo x 2k sin 2 !tx x (- 1 ( ) .. ,. 2 2 j=l [ [ j= l also has a finite limit as t approaches zero. Then this is also true fo� each summand on the right of this equation and we use again Fatou ' s lemma to conclude that the moment of order 2k + 2 exists for the distribution func tion F1 (x) (j = 1 , 2 , . . . , s) . This completes the induction. We prove next that the functions /1 ( t) are analytic characteristic func=
.
•
t�o
t--+0
.£.J lf.. :J
.£.J
rx
_
=
k-1
kJ
.£.J lf.. :J J j
ll.. . 'J
- 00
•
I
dF:J.
295
a -DECOMPOSITI ONS
tions. We raise equation ( 10 . 1 . 1 ) to the power 2k and differentiate it 2k times. We write g ( t) = [f ( t)] 2k and obtain ( 1 0 . 1 . 7) S1 * (t) + S2 * ( t) + Sa * (t) = g( 2k> (t) . Here S1 * (t), S2 * (t) and Sa * (t) contain the same kind of terms which we had in S1 (t), S2 (t) and Sa (t) respectively ; expressions (10. 1 .7) and (10. 1 .3) differ only in the numerical values of the coefficients. This difference is due to the fact that we raised (10.1 . 1 ) to the power 2k. We have then Jj2k> (t) sl * (t) - f(t) . � 2ka.; f ( ) . :J t j= l Let be the distribution function which belongs to g(t) and denote by ex�0> , the algebraic moments of order r of . . . , Fs F1 ex�l > . . . , ex�s> , respectively, and put t = 0 in equation (10.1 .7) . Since s2 * (0) = 0 we obtain ( 10. 1 .8 ) S1 * (O ) + Sa * (O ) = ( 1 ) k ex �l where 8
G(x)
G(x), (x),
(x)
-
( 10. 1 .9 )
sl * (0 ) = ( - 1 ) k � 2kex :i ex�l· j= l We conclude from the fact that we raised ( 10 . 1 . 1 ) to the power 2k that 8
Sa * (O) = ( - 1 ) 'c C where C is a positive constant. It follows then from ( 10. 1 . 8 ) and (10.1 .9) that
( - l)k [S1 * (0 ) + Sa * (0 )] =
so that
� 2kex1 ex�k + C = ex�� •
j= l
(j = 1 , 2, . . . , s ; k = 1 , 2, . . . ) . The number ex�� is the (2k) th moment of the distribution function belonging to [f(t)] 2k, that is, of a distribution function which depends on k. It is therefore not possible to conclude from (10. 1 . 10) and the fact that f ( t) is an analytic characteristic function that the power-series expansion of f1 (z) converges, at least in the circle of convergence of f(z) . Our next aim is to show that the f1 (z) are analytic characteristic functions. Let be the radius of convergence of f(z) ; according to Cauchy ' s integral formula we have 2k d 2k ) f( 2k [ ] ! ) ( z dz = . Cl.. 2(ok> = dt 2k [/( t)] 2k l 2 k+ 2:Jl'l C Z t=O ( 1 0 . 1 . 1 0)
ex �� < ex�o,J
R
where C is the circle I z I =
J
R/2 . Let M0 = sup I f(z) I , then (]
296
CHARACTERISTIC FUNCTIONS
2M0jR, and we see from ( 1 0 . 1 .1 0) that rfw�� < ( 2k) ! Mik (j = 1 , 2, . . . , s) . It follows then that fi (z) = fi (t + iy) is an analytic characteristic function which is regular at least in the strip I Im (z) I < M1 1 • One also sees easily that the fj (z) have no zeros in this strip and that the relation (10 . 1 . 1 ) holds in I Im (z) I < M1 1 • We introduce the functions g (z) = fj ( - iz) = J'" oo dF; (x) (j = 1 , 2, . . . , s) g(z) = f ( - iz) = r' e""' dF(x) oo where z = t + iy (t, y real) . The integrals representing the functions gj (z) converge at least in the circle I z I < M1 1 and the relation ( 10.1 .1 a) 11 [gi (z)] cx1 = g(z) 1 j= holds in this circle ; the function g(z) is regular in the circle I z I < R.
where M1 =
i
lf"'
s
In order to prove that the analytic characteristic functions fi (z) are regular, at least in the strip of regularity of f(z), we must show that the radius of convergence of the series expansion of fi (z) around the origin is We carry the proof indirectly and assume that at least at least equal to It is no restric o ne of the series has a radius of convergence inferior to tion to assume that /1 (z) has the smallest radius of convergence r 1 < Clearly r 1 is also the radius of convergence of g 1 ( z) . We note that the functions gj(z) have non-negative coefficients and conclude from Prings heim's theorem that the point z = r 1 is a singular point of g 1 (z). Let � < r 1 2 be a small positive number and put r/),. = r1 - �. We see from ( 1 0 . 1 . 1 a) that the relation
R.
R.
R.
j
( 1 0. 1 . 1 b)
[g i (rLl + w)] ct1 == g (rLl + w) 11 j= l s
I
is valid for sufficiently small w 1 . The expansion of g j ( rLl + w) according to is powers of w has non-negative coefficients and the coefficient of 1 gi (rLl + w) O �, d n. w= In order to obtain an estimate for this coefficient we raise ( 10 . 1 .1 b) to the power n, differentiate n times, and put w == 0. We can again assume that rfwj > . 1 and therefore obtain a sum of positive terms and conclude that nrfwj (10. 1 . 1 1 ) g(ra) "2. [g(rt. + w w = O g; (ra + w) w = < o ; 1 g; (ra)
wn
dn wn
�
1,
p.
See Titchmarsh (1 939),
3 89.
p.
dn dwn
2 1 4,
or
1-Iille (1 962) , 1 ,
p.
dn dwn
1 33,
or
>J
' n
Markushevich (1 965).
297 The function g(r� + w) is regular for I w I < R - r1 • We put � (R - r1 )/2 and apply Cauchy's integral to estimate the expression on the right of (10.1.11). We see that n n f d n w g(r Ll + )] f [ · n (10.1.12) dw. . dwn [g(rLl + w)] 2nz Let p(rLl) sup I g(rLl + w) I , since (R+ r1 ) /2 > rLl + � we see that p(rLl) C (r1) where C(r1) sup I g(z) I is a positive function of r1 but does not depend on � . It follows then from (10.1.12) that dn [g(rLl+w)]"' � n! [C(r1)] n n! [Ct (rt)]"' 15"' dw"' where C1 (r1 ) is a function of r1 and is independent of �. The terms on the left of (1 0.1.11) are all positive, so that d"',. [g; (rLl + w)]"' /1 � n! gitLl1 [Ct (rt)]"'. d g rLl The radius of convergence of the expansion of g ( rLl + w) according to powers of w is therefore not less than [ C1 (r1 ) ] -1 for any (0, r1 /2). But the function g 1 has a singularity at the point z r 1 , so that the radius of convergence of g1 (rLl w) cannot exceed �. Select � so small that < • The assumption that r1 < R then leads to a contradiction. The [C1 (r1)]of- 1convergence of fj (z) around the origin is therefore at least equal radius to R, so that the characteristic functions fj ( z) are regular, at least in the strip of regularity of f(z) . We have therefore established theorem 10.1.1 in the case where the distributions F(x), F1 ( x), . . . , Fs (x) are symmetric, and must now consider the general case.
a - DECOMP OSITIONS
==
w =O
=
=
wn + 1
!w! =6
lw l =6
0 [respectively for property that lim A(t+iy) I t I < Ll and y < 0]. The class of characteristic functions just described 11�0
=
includes the class of analytic characteristic functions but is more extensive. For the sake of brevity we will call "boundary characteristic functions" those characteristic functions which are boundary values of analytic functions without being analytic characteristic functions. If is a bound ary characteristic function then we can extend its definition to complex = values of the variable by writing
f(z) A(z).
f(t)
The integral representation In this section we derive a number of properties of boundary charac teristic functions which are similar to results for analytic characteristic functions obtained in Chapter We give first a necessary and sufficient condition which a distribution function must satisfy in order that its characteristic function be a boundary characteristic function. 11.1
7.
Theorem 1 1 . 1 . 1 . Let F (x) be a distribution function and f(t) be its charac teristic function. The function f(t) is the boundary value of an analytic function A(z) (z t +iy ; t, y real) which is regular in the rectangle I t I < 0 < y < b if, and only if, the integral f' oo e- •�<JJ dF (x) exists and is finite for 0 y < b but does not exist for y < 0. =
d,
�
We first prove that the condition of the theorem is sufficient and assume therefore that
307
BOUNDARY CHARACTERI STIC FUNCTIONS
y < b. Let g1 (z) = J� eizre dF(x) (1 1 . 1 . 1) g2 (z) = J � e= dF(x). The function g 1 ( ) is regular for lm (z) < b, while g 2 (z) is regular for lm ( ) > 0. Therefore oo z) = g1 (z)+g2 (z) = J oo ei F(x) A( . is regular in the strip 0 < lm (z) < b ; moreover, oo lim A(t+ iy) = J eitz dF ( x) = f ( t). oo To prove the necessity of the condition we assume that f ( z) is a boundary characteristic function. Let g 1 (z) and g2 (z) again be given by (1 1 . 1 . 1 ) , then f(t) = gl (t)+ g2 (t). (1 1 . 1 . 2) The function f( z) is regular in a rectangle, say in D1 = { I t I < �, 0 < y < b }, while g 2 ( ) is regular in the upper half-plane. Therefore g1 (z) is regular in D1 ; on the other hand, it follows from the definition of g1 (z) that it is regular in the lower half-plane, so that g1 (z) is regular in the rectangle D 2 which is symmetric to D1 with respect to the real axis. It follows then from Schwarz ' s symmetry principle ( see Appendix E) that g 1 (z) is regular in the rectangle D = {I t I < �, I y I < b}. We first assume that F(O) 0 and consider the function g1 (t) = 1 J 0 e'"'dF ( ) F(O) F(O) This is an analytic characteristic function whose strip of regu larity con tains the strip I lm (z) I < b. Therefore gl (iy) = J� oo e--1/x dF(x) exists and is finite for I y I < b. Since the integral J� e-""" dF(x) exists for y 0, we see that Joo oo e-11"' dF(x) exists and is finite for 0 y < b, so the necessity of the condition is proved if F( O) 0. In the case where F(O)A criterion = 0 we note that f (t) = g 2 (t) and obtain the necessity of the condition. analogous to the statement of theorem 1 1 . 1 . 1 holds for characteristic functions which are boundary values of functions regular in ct n l {I t I < �, - a < y < 0 } contained in the lower half-plane. e in the proof theorem 1 1 . 1 . 1 indicates
is finite fo r 0
�
00
z
z
""'
d
1/ t O
z
i=
-
't
;;;;:
a
re a g e The argument u s d
X ·
oo
i=
of
�
that
a
308
CHARACTERIST IC FUNCTIONS
boundary characteristic function can be represented by a Fourier Stieltjes integral in a horizontal strip. Let < < be the strip of greatest width in which the boundary characteristic function admits the representation
0 y f3
f(z)
( 11.1.2) f(z) f ' oo eizz dF (x) ( 0 < Im (z) < {J). The strip 0 < lm (z) < {3, in which (11.1. 2) is valid, is called the strip of regularity of the boundary characteristic function f(z) . The validity of the representation ( 11.1. 2) is the reason for the similarity of many properties =
of boundary characteristic functions and of analytic characteristic functions. The discussion of these properties is facilitated by the following lemma :
f(z) be a boundary characteristic function with the strip Let Lemma 11.1.1. of regularity 0 < lm (z) < f3 and choose a real 'YJ such that 0 < 'YJ < f3. Then ) f(z+ i 'YJ (11.1.3) h(z) f(i'YJ) is an analytic characteristic function which is regular in the strip lm (z) < f3 - 'YJ . Lemma 11.1.1 is analogous to lemma 10 .1.1 for analytic characteristic functions and is proved in the same way. The distribution function corresponding to h(z) is (11.1. 4) H(x) � r oo e- 1]11 dF(y), where C f(i'YJ) and F(x) is the distribution function corresponding to f(z). Corollary 1 to theorem 11.1.1. The strip of regularity of a boundary characteristic function f(z) has one or two horizontal boundary lines. One of these is always the real axis. The purely imaginary points on the boundary are singular points off(z). Corollary 2 to theorem 11.1.1. Boundary characteristic functions have the ridge property moreover the zeros and the singular points of boundary characteristic functions are located symmetrically with respect to the imaginary . axzs. Corollary 3 to theorem 11.1.1. A boundary characteristic function has no zeros on the segment of the imaginary axis located in its strip of regularity . Corollary 4 to theorem 11.1.1. Let f(z) be a boundary characteristic unction whose strip of regularity is 0 < lm (z) y < {3. Then log f(iy) is fconvex for 0 < y < =
'YJ
0 and Im (z) > 0 [respectivel y Im (z) < 0] . The extremity of F(x) is given by lext [F] = - lim y - 1 log f(iy) [respectively rext [F] = lim y 1 log f( iy ) ] . The formulae for the extremities were derived in Chapter 7 under the p
R
1.
x� oo
x� oo
y� oo
-
X
X
y� oo
-
assumption thatf(z) is an analytic characteristic function. We now see that they are also valid if this restriction is dropped. To prove theorem we first note that the given distribution and the distribution which is defined by have the same extremities. The statement follows immediately from lemma and from theorem and its corollary.
7.2.2
Remark.
11.1.2 H(x)
(11.1.4)
11.1.1
F(x)
A one-sided distribution function has either an analytic characteristic function or a boundary characteristic function. We next consider factorizations of boundary characteristic functions.
Lemma 11.1.2. Let F(x), F1 (x) and F2 (x) be distribution/unctions and f(t),f1 (t) andf2 (t) their characteristicfunctions. Suppose that f e-11"' dF(x) exists and is finite for 0 � y < b. If F = F1 * F2 then the integrals J e- 11"' dF1(x) (j = 1, 2) exist and are finite for 0 y < b, and the relation 00 f "' e-11"' dF (x) = f e -uo: dF1 (x) f: e- uo: dF2 (x) holds for 0 y oo
oo
oo
�
oo
�
0 and continuous for Im (z) 0. We put for Im ( ) 0 o i _ izu ) r- ( �2 (z) = J oo e zu 1 - 1 + u 2 dM(u); ( 11. 2 .3 ) the function � 2 (z) is regular for Im (z) 0 and continuous for Im (z) 0. Let VJ(z) = Iog f(z) -iaz- � a 2 z 2 -�1 (z). ( 11.2. 4) It follows from our assumptions that VJ (z) is regular for 0 < Im (z) < f3 and continuous for 0 � Im (z) < {3. Since f(t) is infinitely divisible we see from its Levy canonical representation that VJ(t) = � 2 (t) for real t. We conclude then from Schwarz's reflection principle tha t VJ(z) is Im (z) < fl. the analytic continuation of � 2 (z), so that � 2 (z) is regular Therefore �� ' (z) s regular in Im (z) {3. For Im (z) < 0 we have from (11.2.3 ) the integral representation ��' (z) = J=: e'Zl' u2dM(u). a2
oo
a
2
co
�
�
z
0, we see from (1 1 .2. 1 ) that f(t) = exp ( - � a2 t 2) g(t) where g(t) is the infinitely divisible distribution without normal factor which is determined by the constant a and the spectral functions M(u) and oo .
U
o
31 3
BOUNDARY CHARACTERI STIC FUNCTIONS
N(u). This means thatf(t) has a normal factor, contradiction to lemma 11.2.2. (ii) If M (u) is not constant for u < 0, then there exists a finite interval [a, b] , < b < 0 such that C = M(b) - M(a) > 0 . The function g(z) = � exp [ J : (eiz« - 1)dM(u)J is an entire characteristic function which is a factor ofj(z). But g(z) cannot satisfy the conditions of theorem 7 . 2 . 2 , so that g( t) cannot be the charac teristic function of a distribution bounded to the left. This contradicts lemma 11. 2 . 2 , so that the necessity of (ii) is proved. For the proof of the necessity of (iii) we can therefore assume that (i) and ( ii) are valid, and we see from ( 11. 2 .1) that for y > 0 1 3 u log f (iy) = ay+ f (e - uu _ 1 + yu) dN (u) -y f1 1 + u 2 dN(u) + J� (e - uu _ 1 + 1 :uu 2) dN(u). We write l R(y) == J e - vu_y1 + yu dN (u) in
a
o
-
and get
(11.2.5)
o
0
l u3 2dN (u) - J dN(u). logf(iy) ;?;: - ay+yR(y)-y J 1 +u 1 oo
o
We give an indirect proof for (iii) and assume tentatively that
(11.2.6) J : udN(u) = oo. If we can show that (11.2.6) implies (11.2.7) lim R(y) = oo In view of theorem then it follows from ( 1 1.2.5) that lim � Iogf(iy ) = Y 11.1.2, this contradicts the assurrtption that f (t) belongs to a distribution bounded from the left. The necessity of (iii) will therefore be established as soon as we show that (11.2.7) follows from (11. 2 . 6 ). Let H(u) be defined by H (u) = J : vdN (v) (u > 0 ) . 1"'hen H(u) is non-decreasing and we see from (11. 2 . 6 ) that (11. 2 . 8 ) li (O) = Y-+ 00
y� oo
..
oo •
oo .
314
CHARACTERISTIC FUNCTIONS
(e-x - 1 + x)jx is positive and non-decreasing in l u_ e 1 e v + yu l v ;;;;: u R(y) J yu dH (u) J v +v dH( yv) so that R(y) ;;;;: e -1 [H( l )-H(�)J . so that the necessity of (iii) It follows from (11. 2 .8) that lim R(y) = +
It is easily seen that {0, + oo ). Therefore =
o
1
oo,
y-+ 00
is completely proved. We prove next the sufficiency of the conditions (i), (ii) and (iii). We assume that these conditions hold, and we obtain, for real the canonical representation
u i _ log f(t) = iat + J � ( e tu 1 - � 2) dN(u). 1 u
t,
If we replace in the integral the real variable t by the complex variable z = t+ then we obtain a function which is regular in the half-plane > 0 and continuous in � 0. Therefore f(t) is either an analytic characteristic function or a boundary characteristic function, and we see from theorem that the canonical representation is valid in the upper half-plane. Therefore
y
iy
y
11.2 .1 u _ uu a e log f( i y) = - y+ J �( 1 + 1 : u 2) dN(u) (y ;;;;: 0). (11.2 . 9) We put oo u _ l e v Q( y) = J y dN(u) and see from ( 11.2 . 9) that (11.2.10) log f(iy) = - ay+y J 1 + u 2 dN(u)+yQ(y). It follows easily from assumption (iii) of theorem 11.2. 2 that J 1 +u 2 dN(u) < and that lim Q(y) = 0. 0
oo
o
oo
U
U
oo
o
y-+ oo
Therefore we conclude from (1 1 .2. 10) that
iy a - J oo 1 +uu 2 dN(u),
lext [F ] = - lim ! log f( ) = u-+ oo
y
o
so that the proof of the theorem is completed. A theorem concerning infinitely divisible distributions which arc bounded to the right can be stated and proved in the same way.
M I X T U R E S O F D I S T R I B U T I O N F U N C '"f i O N S A N D T RA N S F O RMA T I O N S O F C H A RA C T E R I S T I C FUN CTI O NS 12
In this chapter we discuss briefly certain integral transforms of distribu tion functions. These transformations can be used to construct new characteristic functions from given characteristic functions. Mixtures of distribution functions be n distribution functio11s ; we saw in Let G1 G2 Section that
12.1
(x), (x), . . . , Gn (x) 2.1 F (x) � a1 G1 (x) is also a distribution function, provided that a1 0 (j 1, 2, ) and � a; 1 . We can regard F(x) as a mixture of the distribution functions G1 (x), . . . , Gn (x) with weights ah . . . , an . In the present section we consider a 1nore general mixing procedure and apply it also to the corresponding character istic functions. Let G (x, y) be a family of functions which has the following properties : (i) for each value of y the function G (x, y) is a distribution function . 1n x ·, (ii) G (x, y) is a measurable function of y. The functions G (x, y ) form a family of distribution functions which depends on a parameter y. In the following we consider only families {G (x, y)} of distribution functions which satisfy conditions (i) and (ii). An exhaustive discussion of mixtures of distributions was given by H. Robbins (1948). Let H(y) be an arbitrary distribution function ; we form the expression oo (12.1.1) F(x) f G(x, y)dH(y) and see easily from the dominated convergence theorem [ Loeve (1963) pp. 124- 127] that F(x) is a distribution function. The corresponding characteristic function is then given by (12.1.2) f(t) s:oo g(t, y) dll ( y) =
n
j=l
�
n
j=l
=
=
00
=
=
. . . , n
316
CHARACTERISTIC FUNCTIONS
where
i1"' e d.,
g( y) J :"" G (x, y) . Mixtures of distributions occur in a great variety of practical applications and were used by W. Feller (1943) to study contagious distributions. Two particular cases of (12.1.1) are of some interest. (I) If G (x, y ) is a purely discrete distribution, G(x, y) = � Pv ( y) s(x -�v), then F(x) = � [ J :00 Pv (y)dH(y)] e(x - �v) t,
=
f)
is also a discrete distribution. The distribution which is obtained for
pv { y) = 11 Y" �v = v (v = 0, 1 , 2, . . . ) e
-
v.
-
r,
is called the compound Poisson distribution. ( II ) If
H(y) = � Pv s( y-nv) a purely discrete distribution then ( 1 2. 1 . 1 ) yields F (x) = � Pv G (x, 'fJv)· We put here v A Pv = v . , 'YJv = V and let G(x, ) = [ G(x)]"'* be the v-fold convolution (t) of G(x) with itself ( = 0, 1, 2, . . . ) . Then v A F (x) = v�=O [G (x)]"'* is called the generalized Poisson distribution. Its characteristic function is f(t) = exp {A[g(t)- 1] } . In the following \Ve consider a slightly more general mixture of distribu tion functions. rfhe process still has the form ( 12.1.1) but we relax the assumptions concerning the weight function H(y) in so far as we do not require that H (y) be a distribution function. We will only assume that H (y) is a non-decreasing function whose total variation is e qual to 1. We = 00 J "
is
'V
v
e - .\
v
oo
(t) Thus G (x, v) is defined by G (x, 1 ) G (x, v)
for
v ==
2, 3, . . . .
=
00
,
-
.\ e- -, V.
G (x) and the relations
G (x -y, v - 1 ) dG ( y)
317
MIXTURES AND TRANSFORMATI ONS
- oo) =1= 0 and make no assumptions therefore admit the possibility that concerning the values of at its discontinuity points.
( H H( y) Theorem 12.1.1. The function (12.1.3) F(x) J oo G(x, y)dH (y) is a distribution function whenever {G (x, y) } is a family of distribution functions which has the properties (i ) and ( ) if, and only if, the conditions (a) H (y) is non-decreasing oo 1 (b) J oo dH ( y) are satisfied. We note that the characteristic function of F(x) is (1 2. 1 .4) j(t) J oo 00 g(t, y)d ( y), where g( t, y) is the characteristic function of G (x, y) . The sufficiency of the conditions follows easily from the properties of the family G (x, y) and from the dominated convergence theorem. To prove that conditions (a) and {b) are necessary we specialize the family G (x, y). Let and 'YJ 2 be two arbitrary real numbers such that 1] 1 < 'YJ 2 and select 1) if 'YJ < y 'YJ x s( { G (x, y) s(x) if y or y 1]2• It is then easily seen that F(x) s(x - 1) [H('YJ 2)-H(n1)] +s(x)[1 - H('YJ2)+H (n1)]. If F(x) is a distribution function then we must necessarily have H ('YJ2)- H ('YJt) 0 s(x) for ally and see that which proves (a). To prove (b) we select G (x, y) 00 F(x) e(x) J 00 dH ( y) =
00
ii
=
lJ
=
'Y}l
�
1
=
2
� 'Y} 1
>
=
�
=
=
so that (b) must be satisfied. We remark that condition (a) is necessary only to assure that the weight function should produce a distribution function, whatever family, satis fying (i) and ( ii), is used in (12. 1 .3). We illustrate this by an example where a non-monotone weight function is used to transform a suitably chosen family into a new distribution. Let H ( y)
=
1
\/(2)[:yt(2) - i] [e( y - 1 ) + e(y + 1 ) - V(2) s( y)]
318
CHARACTERISTIC FUNCTIONS
be the weigl1t function and consider the family of normal distributions that is ( t) with mean and standard deviation
y
2a,
) x-y G (x, y) ( 2a . Then F(x) v'(2) [ �(2)- 1 ] [�(x�l ) + � (x:al ) - v'(2)�(;:)] . =
ci>
=
.This is an absolutely continuous function with derivative
F' (x) 2ay'(4n) �v'(2)- 1 ] {exp [- (x - 1)2 /8 a 2] + exp [- (x+ 1) 2 / 8a 2) - y (2) exp [ - x 2 j 8 a 2]}. The function F(x) is a distribution function whenever F' (x) 0 for all x; it is easy to show that this condition is satisfied if a 4 1og1 The characteristic function of F(x) is, according to (12.1. 4), given by f(t) _- [ v(2)v(2)cos-t-1 t ] exp (-2a2 t 2). =
X
�
2
z·
�
Transformations of characteristic functions We have already noted that the mixture of distribution functions in duces a mixture of the corresponding characteristic functions. We now to discuss certain transformations of use the results of Section characteristic functions. 12.2
12.1 Theorem 12.2.1. Let {fv (t)} be an arbitrary sequence of characteristic functions and {av } be sequence of real numbers. The necessary and sufficient condition that (12.2 .1) f(t) � av fv (t) should be a characteristic function for every sequence of characteristic functions is that (12.2 .2) v This follows from theorem 12.1.1 if we put H( y) � av c(y -v). a
=
00
v=O
=O
(t) Here
=
4)(x)
=
1
V 2 77
00
v=O
f ro
_ 00
exp
( z2/2 ) dz. -
3 19
MIXTURES AND TRANSFORMATIONS
g(t) be an arbitrary characteristic function and write fv (t) [g(t)] v v = 0, 1 , 2, . . . . We obtain immediately the following Corollary to theorem 12.2. 1 . Let g( t) be a characteristic function and let A(z) be a function of the complex variable z which is regular in r z I < R, where R > 1 . The function A [g(t)] is also a characteristic function if, and only if, A(z) has a power= series expansion about the origin zoith non-negative coefficients and if A( 1 ) 1 . The corollary can also be derived directly from theorem 12. 1 . 1 if we set H (y) = v av s(y - v) and G(x, y) = [ G(x)] v* for v � y < v + 1 . An interesting generalization of the corollary to theorem 12.2. 1 has Next we let = for corollary :
�
been derived by C. S. Herz ( 1 963) and also by A. G. Konheim-B. Weiss ( 1 965).
Theorem 1 2.2.2. Let =A(z)A[g(t, be a function of the complex variable z which has that f(t) )] is a characteristic function whenever g(t) is the propert y a characteristic function. Then A(z) can be represented by a series, convergent for I z I 1 , which has the fornz � a A(z) = where the arn.n are real and where am.n ;:::: 0 and � � a n = 1 . Clearly the function A(z) need not be regular ; a simple example is the function A(z) = I 1 2 • For the proof of theorem 12.2.2 've refer the reader �
00
rn = O
00
n m,n zrn z
n=O
00
00
m
.
m = O n=O
z
to the papers quoted above. In a subsequent paper A. G. Konheim-B. Weiss (1 968) investigated transformations of non-negative definite functions into infinitely divisible non-negative definite functions.( t) They obtained the following result, formulated here in terms of characteristic functions.
Theorem 12.2.2a. Let O(z) be a complex-valued function of the complex variable The function O(z) has the propert that f(t) = O[g(t)] is an z. y infinitely divisible characteristic function whenever g( t) is a characteristic function if, and only if, O(z) = exp [cA(z) - 1 ] , where c is a positive constant, while A(z) is the function defined in theorenz 12.2.2. The sufficiency of the condition follows from theorem 12.2.2 and De Finetti' s theoretn ; for the proof of its necessity the reader is referred to the paper quoted above. (t) These
Hu thors worked in d1c tnore general framework of locally compact Abelian groups havin g clelnc n t H c)f nrb itru l'i ly high ord er. M
3 20
CHARACTERISTIC FUNCTIONS
Theorem 12.2 .3. Let g(t) be an arbitrary characteristic function and let p be real number such that p > 1 ; then f(t) = pp--g(t)1 is an infinitely divisible characteristic function. Let n be an arbitrary integer and put H(y) = v=O� av s(y- v) a
00
[p - 1 ] 1/n
where
(1 +n)(1 +2n) 1)n . . . [1 ] +(ka ak - 0 ( )k k , for k = 1, 2, . . . and set G (x, y ) = [G (x )] v* if v y v + 1. This shows that 1/ n [ ] p 1 f(t) = p-g(t) is a characteristic function for any positive integer n ; in other words, f ( t) is an infinitely divisible characteristic function. Theorem 12. 2 .3 follows also from the corollary to theorem 12. 2 .1 if we put 1/n 1 ] p [ A(z) = p - z and understand that A(z) is the principal value of this power. We next discuss a few additional transformations. Let V (x) be a non decreasing function of bounded variation defined on the interval [0, M] p
.
wh 1le
_
�
�
.
0, so that g(t) is a mixture of characteristic functions, and the statement follows from theorem 12.1.1. A particular case of some interest is obtained by putting VJ(t) h(t) - 1 where h ( t) is an arbitrary characteristic function. It follows from lemma 5 .4. 1 and theorem 12. 2 .6 that f(t) f [i(1 -h(t))] is a characteristic function , provided that the distribution which belongs to f(t) satisfies the conditions of theorem 12. 2 . 6 . Theorem 12.2.7. Let f(v'iz) be an analytic characteristic function and suppose that the corresponding distribution function F(x) is bounded to the left and that lext [F] 0; then f(iz) is also a characteristic function. Let F(x) be the distribution function of f(v'iz) . Then p*(y) J : 2 � exp ( - r;) aF (x) �
=
=
=
=
a
=
=
=
=
is the frequency function of a mixture of normal frequency functions. Let
oc:
=
J:oo y• p*(y)dy
be the moments of this mixture. Then cx:k- t
=
0
2) 1 y k 2 exp ( - 4 dF (x) dy. y oc2� x J J z v';x We change the variable of integration by putting y u v2x and see, after an elementary computation, that ( 2 k) 4 ( 1 2 .2 . ) cxh cx2k while
=
*
=
2 r 00 r 000 0
k!
!
=
325
MIXTURES AND TRANSFORMATIONS
rxk is the moment of order k of F (x). The characteristic function j(v'iz) has the expansion k i rx k f( Viz) k!
where
-
=
oo
so that
f(u) Therefore
:i"
k'i:.o
� =
.£.J k=O
rxk u 2k
k'
•
.
k ( !( zz .) z and we see from (12.2.4) that k 1 rx ( . ) !( zz ) � (2k) f :k z Since this is the characteristic function which belongs to the frequency function p*(y), the statement is proved. � - 1) = �0 k!
-
(Xk
2k
oo
2k
k O
•
As an example we mention the characteristic function
f(t)
=
. ViZ . . s1n vTz
D. Dugue (1 966) has shown that this is the characteristic function of a distribution which is closely related to the distribution function of Kolmo gorov' s statistic used extensively in the theory of non-parametric statistical tests. We conclude this chapter with the discussion of transformations which are not the result of mixtures of distributions. be an arbitrary characteristic function and denote its distribution Let function by Then
g(t)
G (x). h(u) J "g(y) dy J u J =
=
0
0
00
- oo
eiux
dG (x) dy.
It is easily seen that the order of the two integrations can be exchanged, so that
h(u) J =
We introduce the integral (1 2.2.5) so that
00
eiux _ l
dG(x). zx .
- 00
�(t) - L J >(y) dy du =
cfo(t)
= -
J ' h(u) du J t J 0
=
0
ex:>
1 - ei·uro �
zx.
dG
(x) dtt.
326
CHARACTERISTIC FUNCTIONS
It is again possible to exchange the order of the integrations and one obtains
�(t) J oo (eitx _ 1 - itx) dG(x) x2 . =
-
oo
The last formtda agrees with the Kolmogorov canonical representation (theorem 5 .5 .3) and we conclude that the function as defined by ( 2.2.5 is the logarithm of an infinitely divisible characteristic function which has a finite second moment. We have therefore obtained the follovv ing result :
�( t),
1 ),
Theorem 1 2.2.8. Let g(y) be an arbitrary characteristic function; then f(t) exp { - J : J : g(y) dy du} (12.2.6) is the characteristic function of an infinitely divisible distribution with finite second moment. As an example we consider the characteristic function g( t) e- l tl of the Cauchy distribution. The corresponding function ( 1 2.2.6) is then f1 (t) exp (- I t I+ 1 - e- ltl) ; ( 12.2.7) this is an infinitely divisible characteristic function with finite second moment. The function f2 ( t) exp ( e- l tl - 1) is also ( lemma 5 . 4. 1) an infinitely divisible characteristic function, and we obtain from ( 12.2.7) tl1e =
=
·
=
=
relation
e-ltl f1 (t) f2 (t). This indicates that it is possible to decompose the Cauchy distribution in =
such a way that both factors are infinitely divisible but do not belong to stable distributions.
APPENDIX A The notations 0 and
The notation 0.
o
Let f(x) and g(x) be two functions and assume that g(x) is positive for sufficiently large x. We say that f(x) is at most of the order of g(x) as x tends to infinity an� write f(x) O[ g(x) ] as x -+ oo if there exists a value x 0 and a constant > 0 such that I f(x) f < Ag(x) for x � x0. Thus f(x) O[g(x)] means that the quotient l f(x) 1/g(x) is bounded for sufficiently large x. A. l
A
=
=
Examples. 2 yx
it
O(x), x + 1 0(1), exp ( y'log x) O(x), exp ( x) O(x), 1 jx O(x -312 ), x sin x O(x). In all these examples we have taken for granted that the statement holds as x -+ oo. We write f(x) 0( 1 ) to express that f (x) is bounded as x increases. We list a few rules for the- use of these notations. (I) /1 (x) O[g1 (x) ] , /2 (x) O[g 2 (x) ] imply that /1 (x) +/2 (x) O [g1 (x) +g 2 (x) ] . > 0 is a constant then f(x) O[ag(x)] implies ( II) If f(x) O[g(x)] . (III ) If /1 (x) O[g 2 (x)] then /1 (x) /2 (x) O [ g1 (x) ] a nd /2 (x) O[g1 (x) g 2 (x) ] . y o " o ". Let f(x) and g(x) be both defined and positive for A.2 sufficiently large x. We say that f(x) is of smaller order than g(x) as x ·-+ oo and write f(x) o [g(x)] as x ->- oo if f(x) lim ro� oo g(x) log x o(x), x o(x312 ), etc. We list a few properties of this symbol. (I) f(x) o [g (x) ] implies f(x) O [g(x) ] . o [ g 2 (x)] then /1 (x) /2 (x) (I I ) I f /1 (x) O [g 1 (x)] a nd /2 (x) o[g1 (x) g2 (x)] . We write f(x) o(1 ) to indicate that f(x) tends to zero. =
=
=
=
=
=
=
=
=
=
a
=
=
=
=
=
The s mb l
=
=
Examples.
=
=
=
=
=
=
0.
=
=
328
APPENDICES
The same symbols 0 and o are used if x does not tend to infinity but to some finite value ; it is also possible to use this notation if the variable assumes only integer values as it tends to infinity. This cannot lead to any misunderstanding since the context will always indicate the variable and the limit which it approaches.
APPEN D I X B Schwarz's inequality We prove this inequality in the form in which we need it : namely as an inequality which refers to Lebesgue-Stieltjes integrals with respect to a distribution function. Let F (x) be a distribution and consider two real-valued functions g(x) and h(x) and suppose that g 2 (x) and h 2 (x) are both integrable with respect to F (x) over ( - oo, + oo) . Then
oo
J oo [u g(x) + v h(x)) 2 dF (x)
is a non-negative quadratic form in the variables u and v, so that the discriminant of this form is non-negative. This yields 2 [h(x)) 2 dF (x) g(x} h(x) dF (x) � [g(x)) 2 dF (x)
J J 00 which is the desired inequality. [ J : oo
J: oo
00
AP P E N D I X C Weierstrass' approximation theorem We need here only the trigonometric approximation theorem of Weierstrass and introduce the following notation. We denote by the class of all continuous functions f(x) , defined for all real x , which are periodic with period We define trigonometric polynomials Tn (x) of period 2n and degree n
CL
L.
� n
( cxv cos vx + ,Bv sin vx). v =O It is sometimes convenient to write these in complex form as +n Tn (x ) 2: av eivx V= - n Tn (x)
=
=
329
APPENDICES
av
Tn (x)
where the can easily be expressed in terms of the cxv and f3v· If is a trigonometric polynomial with period 2n then (2n / ) is a trigono metric polynomial with period
Tn x L L. eierstrass' (trigonometric) approximation theorem. Let f(x) c2T&; then W for every e > 0 there ·exists a trigonometric polynomial Tn (x) (of period 2n) such that I f(x) - Tn (x) I < e for all real x. If f( x) CL then f [Lyj(2n)] C2n so that one obtains essentially the same appro ximation theorem for the functions of C the approximating trigonometric polynomials then have necessarily the period L. E
E
E
L;
A convenient proof of the theorem may be found in I. P. Natanson (1955) [see § 2] or in N. I. Achieser (1956) [see § 22] .
AP P E N D I X D Order and type of entire functions
Let f (z)
=
� ck zk be an entire function. 00
k=O
We denote by f) (D. 1 )
M(r ;
=
I
max f(z) I lzl 1 . Since k(z) = z- 2 J: (z - u)eudu+iz - 2 e"' J: (z - x - iv)eivd�', ==
o
=
.
1
we see that Therefore
l k(z) l � 1 + 2 exp [Re (z)] for all z. Since 3 zu zu ezu - 1 - 1 + u 2 = (zu) 2 k(zu) 1 +u 2 (E. 1 . 1 ) � 2u 2 { 1 + eu < z) ) { I 1 2 + I z I) , we conclude that the assumption Ja u 2 dN(u) < oo implies +
.Re
-l- 0
--
Z
th e uniform
33 1
APPEND ICES
convergence of the integral defining f(z) on every bounded z-set. The function f(z) is therefore an entire function. We see from (E. l . l ) that I f(z) I �
4 j z j 2 { 1 +ea Re (z)) J a 2 a 2 2 {) 81 1 J z
+o
u
dN
u
+o
u
dN ( u
)
1 and Re (z) > 0 if I z I > 1 and Re ( z) � 0 . if l z l >
The estin1ate of the lemma follows immediately.
Proof of lemmas 9.1.2A and 9. 1 . 2B exp ei z ) where the coefficients satisfy the estimate Letf(z) = }:. (E.2. 1 ) dP = 0 [exp ( - kp 2 )] ( P --+ oo ) . E.2
p O
dp
Condition (E.2. 1 ) ensures that the series for f(z) converges absolutely and uniformly in every bounded set of the z-plane. Therefore f(z) is an entire function which obviously has the period i T. We write Re (z) and see that
x= npx npx Z Z }:. { 2 e exp xp f(z) ( -kp + T )} I � }:. 1 I I ( T )=o 2 x = o{ex; (�� 2 t }:. ;xp [ - k ( p - YJ } :; = o [exp (��:)]. 2 x n ] The last estimate follows from the fact that � exp [ - k ( p is ) kT a continuous periodic function of x with real period kTjn and is therefore bounded. If x = Re (z) � 0 we see that I f(z) I � � I dP I = 0(1 ) and p =O
dv
p=O
p = - oo
00
p =O
lemma 9. 1 .2A is proved. We proceed to the proof of lemma 9. 1 .2B. We assume that the entire function f(z) is periodic with period i T and that the estimates (9 7) hold. We expand f( iy) into a Fourier series,
.1. f(iy) =
v
� oo dv exp oo
2ni p y ( T ) ( - oo < y < oo)
where
T 2ni p 1 y (E. 2 .2) dP = .. J / { y) exp ( - T ) dy (p = 0, + 1, + 2, . . . ) . 1 l_.�et be complex variable and consider the entire function A( C) = f(iC) ( � 2i'JTjJC/1., ). [Iror (1�.2.2).] t
•
C cxp
a
0
real C this is the integrand in
We
332
APPENDICES
integrate C) around the rectangle whose vertices are the four points 0, iO, iO + and T ( 0 real). According to Cauchy ' s theorem this integral is zero. Since the function is periodic with period T, the integrals along the vertical sides of the rectangle cancel, and we see that 2 P f( iy - 0 ) exp d� = ( y + iO) dy.
A(T
A( C)
� J� It follows that ( 2npO) p ex d (E.2.3) I 11 I �
[ �
]
max f(iy - 0 ).
T o 0 ; we see then from (9. 1 .7) that I d� I � exp
[� 0 + O(log o)J .
We let 0 tend to + oo and see that d1J from (E.2.3) and (9. 1 .7) the estimate d� We put 0 where k
=
=
-
-
=
=
0 for p < 0. If () < 0 we obtain
[ ei 0 + N0 2)] .
o exp
:/r and get the desired estimate d11
n; 2
=
O( - kp 2 )
• 2 NT
Proof of lemma 9.2. 3 Let z = t + iy (t, y real) ; we show first that the function O(z) = (z + 1 ) - c e -bz f(z) is bounded in the half-plane Re (z) � 0. We select a point z0 = t0 + iy0 in the first quadrant ; i.e. z 0 E [ (t, y) : t > 0, y > 0] . Let � e be the angle formed by the rays t = [(t, y ) : y = 0, t � 0] and
E. 3
t. =
[(t, y) : y �;. t 0J . We select e > 0 so that z0 =
�
E
�
•.
We put Oe (z) = O(z) exp ( isz 2 ), then for z E �e we have I O e (z) l = I O(z) l exp ( - 2sty) � M3 exp (dt 2 - 2sty) � M3 exp (dl z l 2). M oreover, I Oe (z) I � I O(z) I � M2 on the ray t and I Oe (z) I � M3 on the ray te . We can therefore apply lemma 8. 2 . 1 with � = �e M = ' n d max { M2 , M3 ), f3 = 2 > ex = arctan � and p = 2 < nj{J . 2 Then I Oe (z) I � M for z E �e and max (M2 , M3 ) exp (2st0 y0). I O(z0) I
�
333
APPENDICES
We let
tend to zero and see that I O(z 0) I � max (M2 , M3). The point z0 is an arbitrary point of the first quadrant, so that O(z) is bounded in the closed first quadrant. In a similar way one shows that t � 0, � 0] O(z) is also bounded in the closed fourth quadrant and therefore in the half-plane Re (z) � 0. We consider the function f(z) = (z + 1 )c- a o (z). 01 (z) = (z + 1 ) - a On the boundary of the half-plane Re (z) � 0 we have l 01 (z) I � M1 while in the half-plane Re (z) � 0 01 (z) = O[j Z + 1 l e- a] . The assumptions of lemma 8.2. 1 are again satisfied if we identify � with the half-plane Re (z) � 0 and put f(z) = 01 (z), M = Mh ex = = and p = � - We see then that for Re (z) � 0 01 (z) I � Mh or I f(z) I � M1 ! z + 1 1 a exp [b Re (z)] , as stated in lemma 9.2. 3 . s
[(t, y):
e - bz
f3 n/2
y
- n/2,
I
lemma
9.2.4 Proof of Let f1 {z) = f(z + iT ) -f (z) ; the function f1 (z) is entire and (E.4. 1 ) l f1 (z) l � exp {k Re (z) + O(log l z l ) } [Re (z) � 0] . It follows from the representation of f(z) that E.4
/1
(iy) . � =
j = - oo
( iTb;) exp
(2nTijy) .
According to the assumption of the lemma, this series converges uniformly on the interval 0 � � T and is therefore the Fourier series of its sum f( iy). We repeat the reasoning which led from formula (E.2.2) of Appendix E.2 to (E.2.3) and see that 2 0 max f f1 exp J 0) 1 (E.4.2) I iTb3 �
y
(�)
(iy -
o w = [l� T/{2n)] . ( 1!11 )
'ro show th is
we
consider inH tcn
,�.
Table of discrete d . Table of absolutely continuous d. Cantor d . Purely singular, strictly increasing d. d.f. having mon�ents of order inferior to m but not of order m or of higher order. Table of moments. Recurrence relations for moments of singular d. (ref. to) two different d. f. having the same sequence of mon�ents. Table of c.£. c. f. of Cantor d. Behaviour of L = lim sup f(t) for singular d.
6 7
1 .2 1 .2 1 .2 1 .4
-
Description of example
Page
___. _
-- --
...... ...... .... . ,
I I
ltl� oo
c. f. for which f'(O) exists but not the first moment. c.f. which is nowhere differentiable. A c. f. which has an e xp ansion f(t) = 1 + o(t), a lthough the first moment does not exist. The weak converge nce of a sequence {Fn} of d. f. to a d. f. F(x) does not imply the convergence of the sequence of moments of the Fn to the moments of F. The weak convergence of a sequence {Fn} of d. f. to a limiting d. F(x) does not imply the convergence of the cor responding densities. Symmetric Bernoulli convolutions exhibiting different be haviour of L = lim sup /( t)l . c. f. of singular d. with L > 0 1 t!4- oo or L = 0 . An ab s olutely continuous d. may have a c. f. which is not abs olutel y integrable . The c. f. of two different d. can agree over a finite interval. Multiple factorizations of c. f. Cancellation law invalid in arithmetic of d. f. A c. f. which has no re al zeros need not be i.d. The absolute valu e of a c.f. is not nece s sarily a c. f. Table of canonica l rep resenta tions of c. f. Two different c. f. can have the same absolute value. Two different c f. can have the same square.
I
.
_
Abbrcviutions used :
= characteristic fu nction (s), d . = distribution (s), d . f. dis tribu t ion functiotl (R), i . d . = infinitely c.liviAib le. c. f.
&:JI
336
LIST O F EXAMPLES
Chapter and Section
5.5 5.5 5.5 5.8 6.2 6.2 6.3 6.3 6.3 6.3 6.3 6.3 6.3 7.1 7.2
7.2 7.2 7.3 7.4 7.4 8.2 8.4 8 .4 9.2 12.1 1 2 .2 1 2 .2
Page
Description of example
A c.f. f( t) such that f( t) is not i.d. but I f( t) I is i. d. An i.d. c.f. may be the product of two factors which are not i.d. 1 23-4 An i.d. c. f. which has an indecomposable factor (factor explicitly given) . A stable density with exponent a = ! . 1 43 An i . d. c.f. can be the product of a denumerable number of 1 76 indecomposable c.f. i.d. c.f. which have indecomposable factors. 1 79 1 83-4 c. f. of an absolutely continuous and unbounded indecom posable d. Product of two c.f. , neither of which has a normal component, 1 84 can have a normal factor. Product of two c.f. , neither of which has a Poissonian com 1 85 ponent, can have a Poissonian factor. Construction of a c. f. which belongs to a finite and inde 1 88 composable absolutely continuous d. Construction of a c. f. which belongs to a finite, purely singu 188 lar, indecomposable d. Factorization of rectangular d. (it can be represented in two 1 89 ways as an infinite product of indecomposable factors) . Factorization of rectangular d. into a product of two purely 1 89 singular d. Quotient of two c. f. need not be a c. f. 1 94 A d. which has moments of all orders but has not an analytic 1 98 c. f. Nevertheless the sequence of moments determines this d. completely. A one-sided distribution may have an entire c. f. of order 203 greater than 1 . The infinitely many zeros of the c. f. of a finite d. t:leed not be 203 real. References to examples of rational c. f. 212 A periodic c. f. which is not analytic. 225 A doubly periodic c. £. 227 Convolution of three Poisson type d. can have an indecom 252 posable factor. i.d. c.f. with zeros on the boundary of its strip of regularity. 258 Entire c. f. without zeros which is not i. d. 259 Reference to an example which shows that a c. f. can belong 280 to !l! but not to I0 • Mixture using a weight function which is not monotone . 317 320-3 Examples of transformations of c. f. Examples of transformations applied to exp ( - t2 /2 ) which 323 yield entire c.f. of order 2 with a finite number of zeros.
1 22 1 22
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I N D EX* Abel summability, 41 absolute moment, 1 0, 25 absolutely continuous, 4, 66 - - d . ' 4, 5, 38, 7 5 - monotone, 302 ACHIESER, N. I . , 329, 3 37 adjoint polynomial, 228 AKUTOWICZ, E. J., 89, 3 37 algebraic moment, 1 0 almost periodic, 1 9, 36 ex-decomposition, 292, 301 -, denumerable, 292 -, finite, 292 - for i.d.d. , 305 - - lattice d. , 302 - - normal d. , 302 - - Poisson d. , 303 - of a. c. f. , 292, 301 , 302 analytic c.f. , 1 91 ff. - -, boundary values of, 306 - -, convexity property, 1 96, 1 97 - -, factorization of, 2 36 - -, i. d. , 25 8 ff. approximation theorem of Weierstrass, 28, 1 02, 328 arithmetic of d.f. , 103, 1 04, 1 66, 1 82 :ff. , 252 asymptotic expansions of stable densi ties, 1 48
boundary c. f. , ridge property of, 308 - -, strip of regularity of, 308, 3 1 0 - value of analytic f. , 306 bounded d. (to left, right), 1 42 - to (left, right), 1 42 - variation, 320 (C, 1 )-summability, 41 CAIROLI, R. , 224, 3 37 cancellation law, 1 22 canonical product, 222 - representation, 1 1 3 :ff. - -, Kolmogorov, 1 1 9 - -, Levy, 1 1 8, 1 76, 1 78-1 79 - -, Levy-Khinchine, 1 1 7 - - of stable c.f. , 1 32 - -, table of, 1 21 Cantor' s ternary set, 8 Cauchy d. , 7, 8, 1 2, 1 8, 85, 91 , 1 08, 1 21 , 1 37, 1 39, 3 26 Characteristic function, 1 1 , 1 5 - -, analytic, 1 91 - -, boundary, 306 :ff. - -, decomposable, 1 0 3 , 238, 254 - -, indecomposable, 1 0 3 , 1 09, 1 79 - -, i. d. , 1 07, 1 08, 1 09, 1 21 , 1 22 - -, periodic, 225 - -, table of, 1 8 - -, transformation of, 3 1 5 , 3 1 8 characteristic, second ( second c.f. ) , 26 , 253 characterization, 254 - of normal d . , 254 - of normal, Poisson, conjugate Poisson d., 256 Chebyshev polynomials, 21 6 chi square d. , 8 Choquet ' s theorem, 1 1 8 CHRISTENSEN, I. F. , 224, 3 37 CHUNG, K. L. , 99, 1 64, 3 3 7 closed limit inferior, 62 closure theorems, 1 1 1 , 1 1 2 complex conjugate, 1 5 components of a d.f. , 3 8 =
BERGSTROM, H. , 1 42, 1 50, 3 37 Bernoulli convolution, symmetric, 64, 66, 67 Beta d. , 7, 8 , 1 8 binomial d. , 6, 1 4, 1 8 BLUM, J. R. , 1 24, 3 37 BoAs, R. P. , 25 , 3 37 BocHNER, S. , 71 , 210, 3 37 Bochner' s theorem , 71 , 79 BOHR, H. , 1 9, 36, 3 3 7 BOREL, E. 222 ' boundary c.f. , 306 - -, extremities of, 309 - -, factorization of, 309
* The following abbreviations are used in the index : c .f.
characteristic function ; analytic characteristic function ; d distribution ; d.f. distribution functions ; a.c.f. f. function ; i.d. infinitely divisible ; i . d . d . infinitely divisible distribution. =
=
=
=
=
=
=
INDEX
composition, 3 7 concave, 91 conjugate complex, 1 5 - d. , 29, 30, 1 6 8 continuity interval, 2 - point, 2 , 29 - theorem, 47, 48, 88, 1 02 - - for bounded non-decreasing functions, 52 - -, second version of, 5 3 continuous, 1 25 -, absolutely, 4, 1 25 - part of d.f. , 3 - singular, 1 25 - (to the right), 2 convergence in the mean, 76 -, radius of, 1 9 1 -, uniform (of a sequence of c.f.), 50 -, weak, 43, 44, 48 convergence theorem, 237, 3 1 5 , 3 1 7 - -, dominated, 3 1 5 , 3 1 7 - --, monotone, 23 7 convergent sequence of d.f. , 42, 43 convex, 83, 9 1 - function, 202 convexity properties of a.c. f. , 1 96, 1 97 convolution, 37, 41 - of normal and Poisson d. , 245 -, symmetric Bernoulli, 64, 66, 67 - theorem, 36 -, v-fold, 3 1 6 convolutions, infinite, 5 5-67 -, -, convergence criteria for, 59-6 1 -, -, convergence of, 59 COPSON, E. T. , 203, 221 , 3 37 CRAMER, H. , 1 , 1 1 , 1 3, 1 4, 21 , 37, 75, 1 69, 210, 243 , 292, 3 37 Cramer ' s criterion, 73, 2 1 0 - theorem, 243, 245 , 254, 256, 302 - -, converse of, 254 criteria for c. f. , 68, 210 - - convergence of infinite convolutions, 59-6 1 CRUM, M . M. , 91 , 3 3 7 cumulant generating function, 26 cumulants, 26 CUPPENS, R. , 289, 300, 305, 3 3 7 CZERNIN, K. E. , 1 5 8 Darboux sums, 37, 1 1 2, 236, 237 decomposable c. f. , 103, 238, 254 decomposition of d. , 3 - theorem, 1 69, 1 70 - - , general , 1 69 De Finetti' s theorem, 1 1 2 degenerate d. , 5,,_, 1 8, 1 9, 1 08, 1 2 1 , 1 66 degree of freedom , 8
345 densities, stable, 148 density function, 5, 3 3 denumerable ex-decomposition, 292 derivative, symmetric, 25 differential equations, 227, 228 - -, c. f. as solutions of, 228 - --, positive definite, 228 Dirichlet conditions, 86 discontinuity point!' 2 discontinuous part of a d. f. , 3 discrete part, 3, 4 distribution : Beta, 7, 8 , 1 8 binomial, 6 , 1 4, 1 8 Cauchy, 7, 8, 1 2, 1 8, 85, 91 , 1 08, 1 21 , 1 37, 143, 3 26 chi square, 8 compound Poisson, 3 1 6 degenerate, 5 , 1 8, 1 9, 108, 1 21 , 1 66 exponential, 8, 1 3 Gamma, 7, 1 3, 1 8, 1 08, 1 09, 1 20, 1 21 , 1 79, 1 91 Gaussian (Gauss-Laplace), 7 generalized Poisson, 3 1 6 geometric, 6 , 1 1 3, 1 76 hypergeometric, 6 Laplace, 7, 1 3 , 1 8, 88, 1 09, 1 22 negative binomial, 6, 1 4, 1 8, 1 03 , 1 21 Neyman type A, 221 normal, 7, 1 3 , 1 8, 91 , 1 08 , 1 21 , 1 3 6, 143, 1 91 , 21 3, 243 , 254, 25 5 Pascal, 6 Poisson, 6, 1 4, 1 8 , 1 08, 1 21 , 1 8 5 , 1 9 1 , 22 1 , 227, 243 rectangular, 7, 8, 1 09, 1 89 Simpson ' s, 7 Student ' s, 7, 1 2, 1 3 triangular, 7 uniform, 7 distribution : bounded, 142, 201 , 202 conjugate, 29, 30, 1 6 8 fin�e, 142, 1 81 , 200, 228, 259 i. d. , 1 03 lattice, 6 , 1 7, 225 , 245 , 302 one-sided, 1 42, 200-202, 309 stable, 1 28 :ff. symmetric, 30, 292 unimodal, 91-99 distribution function : - -, absolutely continuous, 6, 39, 75 - -, discrete, 5, 6, 36, 41 - --, indecomposable, 1 81 , 1 83 - - of a.c. f. , 1 97 ff. - -, singular, 8, 9, 1 9 dominated convergence th eorem, 3 1 5 , 317
346
INDEX
DU GUE , D . , 86, 87, 302, 325, 3 37, 3 38
elliptic function, 227 energy parameter, 263 entire characteristic function, 1 95 , 1 9 8, 206 , 209, 210, 2 39 - - -, determination by properties of factors, 253 :ff. - -, factorization of, 23 8 - - - of infinite order, 210 - - - - finite order and minimal type, 209 - - - - - - - maximal type, 209 - - -, order of, 1 9 5 , 1 98, 20 5-210 - - -, type of, 205-21 0 entire function, 1 9 5, 329 :ff. equivalence (of c. f. or second c. f.), 25 3 EssEEN, C. G. , 20, 54, 3 3 8 EvANs, G. C . , 1 4, 3 38 exponent of convergence, 222 - - stable d., 1 3 3, 1 42, 1 47, 148 exponential d. , 8, 1 3 - type, entire function of, 203 extension of factorization theorems for a. c. f. , 292 ff. - - non-negative definite functions, 89-9 1 - - - - - indeterminate 90 ' ' - - - - - unique 90 ' ' extremities , formulae for, 202, 309 - of d.f. with boundary c.f. , 309 extremity (left, right) of a d.f. , 142 -
Faa di Bruno's formula, 27 factor-closed family, 245 factor, indecomposable, 1 03, 1 70, 1 76 - of a.c.f. , 23 6 - - c.f. , 1 03 factorization of a.c.f. , 236 ff. - - boundary c.f. , 3 1 0 - - entire c.f. of finite order, 239 - - i.d. a.c.f. , 258 :ff. - problems , 1 03 , 1 22, 1 67, 1 9 1 , 236 ff. - theorem, Hadamard ' s, 203 , 222 , 243 , 244 - -, Weierstrass ' , 88 Faltung, 37 Fatou ' s lemma, 229, 293 , 294 FELLER, vv. , 1 22, 142, 3 1 6, 3 3 8 Finetti ' s theorem, 1 1 2 finite ex-decompositions , 292 - d. , 1 42, 1 8 1 , 202 , 228 , 25 9 FISZ, M. , 1 2 5, 3 3 8 Fourier coef-ficients , 36, 86 integral> 1 93 , 1 94 - inversion theorem , 84 - series , 86 -
·
Fourier transforms, 1 1 freedom, degree of, 8 frequency, Poisson, 263 - functions of stable d . , 1 3 8, 147 - -, stable, 1 3 8, 1 47 function, \Veicrstrass ' , 23 - of bounded variation, 320 Gamma d . , 7, 1 3, 1 8, 1 08, 1 09, 1 20, 1 2 1 , 1 79, 1 91 Gauss-Laplace (law of), 7 generating fw1.ction, cumulant, 26 - -, moment, 10, 1 96 , 1 97, 251 - -, probability, 1 0 , 1 82 genus , 21 2 geometric d. , 6, 1 1 3, 1 76 GIL-PELAEZ, J . , 3 3 , 3 3 8 GIRAULT, � . , 1 9, 20, 87, 227, 321 , 3 38 GNEDEN KO, B. v. , 54, 1 04, 1 28, 1 36 , 1 37, 143, 1 62, 1 6 5 , 3 38 GoLDBERG, A. A. , 280, 3 38 HADAMARD, }. , 1 8 3 Hadamard ' s factorization theorem, 20 3, 222, 243 , 244 HAHN, H. , 1 0 1 , 3 3 8 HALMOS, P. R. , 62, 3 3 8 HARDY, G. H . , 23, 8 3 , 3 3 8 HARTMANN, P . , 1 24, 1 25 , 3 3 8 HAUSDORFF, F., 62, 3 3 8 Reily' s first theorem, 44 , 49, 72, 1 73 - second theoren1., 45, 72, 82, 1 1 6 - - -, extension of, 47, 49, 72 , 1 1 6 Hermite polynomials, 78, 323 Hermitian., 71 , 77 HERZ, c. s . , 3 1 9, 338 HILLE, E., 3 30 , 3 34, 338 HINCIN, see KHINCHINE HoBsoN, E. W. , 2, 1 40, 3 3 9 hypergeometric d. , 6
I 0 , 266, 280, 28 1 ff. -, necessary conditions for member ship, 280, 281 -, sufficient conditions for mcrnber ship, 266, 281 ff. IBRAGIMOV, I . A. , 99, 1 5 0, 1 5 8 , 1 6 5 , 258, 339 imaginary part (Im) , 1 92 increase, point of, 2 indecomposable absolutely continuous d.f. , 1 8 3 - c.f. , 1 03, 1 09, 1 79, 1 8 1 , 252 - d. , 1 8 1 , 1 82, 1 83 - factor, 103, 1 70, 1 76 independent, ration a1ly, 249, 287 infin ite convolut ion , 5 5 ff.
347
INDEX
infinitely diviM iblc a . c . f. , 2 5 H fi. - - boundary c . f. , 3 1 0 f l'. - - c . f. ,
