On moment sequences and infinitely divisible sequences

On moment sequences and infinitely divisible sequences

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Hansen, B. G. (1986). On moment sequences and infinitely divisible sequences. (Memorandum COSOR; Vol. 8611). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1986

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EINDHOVEN OF TECHNOLOGY Faculty of Mathematics and Computing Science

Memorandum COSOR 86 - 11 On moment sequences and infinitely

divisible sequences by

B.G. Hansen

Eindhoven, the Netherlands September 1986

On

Moment Sequences and Infinitely Divisible Sequences

B.G.Hansen

Technological University. Eindhoven

ABSTRACT

Two sequences related by the recurrence equations defining infinite divisible lattice distributions are studied. Necessary and sufficient condi-tions are found for either of the two sequences to be one of the three classi-cal moment sequences. These results are similar to those obtained by R.A. Horn for sequences related by the renewal equations.

1. Introduction

Kaluza [7] studied the relationship between two sequences {un ):'=0 and Vn

1:'=0

related by the renewal equations

n

Un+1=

1:

It un-/;' •

n

=

0.1.2 ... uo= 1. (1.1)

1:=0

He established sufficient conditions for the two sequences {un 1:'=0 and Vn 1:'=0 to be posi-tive. showed that {un ):';0 is a Stieltjes Moment Sequence if and only if {In 1:'=0 is and proved that if {un

In==o

is completely monotone. then so is

Itn

}:'=o·

In the theory of infinitely divisible lattice random variables concentrated on the non-negative integers a similar set of equations occurs. namely through the following theorem. which was first proved by Katti [8] and re-proved and exploited by Steutel [11]. Theorem 1.1. A lattice distribution ip" ):'=0 with Po>O is infinitely divisible

il

and only

il

there exists

Ir"

1""":0 such that n

(n +1)PIl +1

=

1:

rl p" -L •

k={J

00

n

=

0.1.2 .... with r" ~ 0 and. necessarily.

1:

rn I(n + 1)

<

00

11=0

(1.2)

Even though the relationship between the generating functions of sequences related by (1.1) and (1.2) are quite different. many results concerning (].1) found by Kaluza and authors after him [2.4.13] have close analogues in terms of (1.2). For example. Kaluza proved for sequences related by (1.1) and Warde and Katti [13] for sequences related by (1.2). that a sufficient condition for the sequence

lin

}:'=o

resp. {rn }:'=o to be non-negative is that {un }:,= 0 resp. {Pn

In==

0 is logconvex and non-negative.

Horn [6] proved for sequences related by (1.1). that {u" 1:'=0 is one of the three classi-cal moment sequences jf and only if

It"

1:'=0

is a classical moment sequence of the same type as {u"

1:':0'

In this paper the moment behavior of two sequences related by (1.2) is investigated. We obtain results equivalent to those found by Horn. We use a method simi-lar to Horn·s. but since the relationship between two sequences related by (1.2) turns out to be more "delicate" than the relationship between two sequences related by (1.1), a more

2

-2. Preliminaries

In what follows the index n of an infinite sequence will always range over the non-negative integers 7L +. The real-valued sequences {Pn } and {rn } will always be assumed to

be related by (1.2), with the added restriction that Po= 1. We shall denote the generating functions (g.f.) of the sequences {p" } and {T·_{n }} by P and R, respectively. We shall also

define R' by

CX)

R'(x)=

L

(rn /(n+1))x n_{+1 .}

n=O

By taking g.f.'s on both sides of (1.2) we get the following formal relationship between P

and R' ;

P(x) = exp{R' (x )} . (2.1)

Thus, if either P or R' has a positive radius of convergence, then so does the other. If this is the case then both P and R' will converge uniformly in some neighbourhood of zero (not necessarily the same neighbourhood).

Let {an} be a sequence of real numbers and let k

=

°

or 1. The sequence {an +k } is

called a Hamburger MOTTl£nt Sequence, and we write {an +k }EH (JR), if and only if there

exists a finite measure J1.~ such that

CX)

-ex:>

If J1./.: can be taken to have support in [0,(0) the sequence is called a Stieltjes MOTTl£nt Sequence, and we write {~H }EH (JR +). If J1./.: has support in [0.1], the sequence is called a

Hausdorff MOTTl£nt Sequence, and we write {an +k )EH ([0,1]). A measure J1. on a cr-field (3 is

said to be bounded by a measure v on (3 if for any BE (3

J1.(B) ~ v(B) .

Sinse the representing measure of a Hamburger Moment Sequence or a Stieltjes Moment Sequence need not be unique, we shall say that such a measure is bounded by a measure v

if there is a measure, in the class of representing measures that is bounded by v.

The following six lemmas, which will be needed in the following sections, state vari-ous properties of and relations between moment sequences. The proofs of the first four lemmas may be found in the references indicated.

Lemma 2.2 [9]. {an)EH ([0.1]) if and only if {an )EH (JR +) and {an) is bounded.

Lemma 2.3 [6]. For each T ER+ let {an (T))EHCi), for i

=

R ,R+ or [0,11. Suppose an := lim an (T )

T-oo

exists for all n E 7L +, then {an) EH

Ci )

for i

=

R ,R + resp. [0,1]'

Lemma 2.4 [9]. If a representing measure of a Moment Sequence has bounded support.

then the representing measure is unique.

Lemma 2.5. {an }EH (JR) if and only if there exists two sequences _{{bn }, {en }EH (JR} +)

3

-Proof:

= 00

f tndJ.L(t)= f tTld J.L(t)+(-t)" ftnd(-J.L(-t)).1

- 0 0 0 0

Lemma 2.6. For each TER let (an (T))EH(i). for i

=

R .R+ or [0.1]. with its represent-ing measure bounded by a measure v. Suppose

an := lim an (T)

T-DC

exists for all n E ~ +. then {an

lEH

(i). for i

=

R . R + resp. [0,1]. with its representing

measure bounded by the measure v.

Proof: The proof for Hausdorff Moment Sequences follows from lemma 2.4 and Helly's first and second theorems. As the proofs for Hamburger Moment Sequences and Stieltjes Moment Sequences are identical. only the former will be carried out.

Let J.Lr be a representing measure of {an (T)} bounded by v. By Helly's first theorem there exists a subsequence {J.LT

k

It'=o

converging weakly to a measure J.L, necessarily

bounded by v. Since {an (T_{t )}} converges. as k - 00. for any n, then for every fixed even N E~ + there exists an M ER+ such that for any k E~ +. any A eR+ and any n ~ N

=

M> f It 1·1\' dJ.LT/t)~ A,'V-n

r

It III dJ.LTk(t).

-ex> It

r'>A

Hence for every €

>

°

we can find an A such that for all k and all n ~ N

r

I tin d J.Lr

k (t) < € .

It

r'>A

By Helly's second theorem. for all n ~ N

.4 A

-1

tn dJ.LT/d-

-1

t

n

dJ.L(r).

Hence for every €

>

°

there exists an A e R + such that for all n ~ N

.4 A A

f til d J.L(t )-a" ~ _f t n d J.L(t)-

J

t 11 d J.L Tit (t )

-A -.4 -.4 A co

+

f t"dJ.LT/t)- f til d J.LT/t ) -A - 0 0 00

+

J

tn dJ.LTt(t)-an

<

E. - 0 0

Thus J.L is a representing measure for {an} and J.L is bounded by

v.1

For 0: E Ii. +. we shall denote by H ",(lR) and H ",(lR +). respectively. the classes of Hamburger Moment Sequences and Stieltjes Moment Sequences whose representing meas-ures may be taken to have support in [-0: .0:]. For 0: E R +. we shall denote by M (0:) the set of real analytic functions on ( - 0 -1,0:-1) having an analytic continuation onto the

- 4

into itself. We shall denote by M' (a) the subset of functions in M (ex) mapping the upper half-plane into the strip,

lz

E(l 10< Im(z) <11-}. H~ (lR) and H~ (R+) are defined as the subclasses of H ,,(jR) and H ",(lR +) resp., whose representing measures are bounded by Lebesgue measure. Likewise. H'CR). H'(lR+), and H'([O,lD are defined as the sub-classes of H (lR). H (lR +). resp. H ([0.1]). whose representing measures are bounded by Lebesgue measure.

From Horn [6] we have the following two lemmas.

Lemma 2.7 [6]. Let 0: e R + and let A be a real valued function defined on (-0: ,0:). The following are equivalent;

(j) A

ex

)E M (a )

(ij) A (x)

=

Lall xn with x eC-a-1,a- 1) and {all +lIEH o.(R)

(iii) there exists a finite measure fJ. on [-0:-1 ,a-I] such that

A(x)=A(O)+

J

~dfJ.(r).

[-0.,,,,) -xt

Lemma 2.8 [6]. Let a E R + and let {a" I be a sequence of real numbers such that

A (x)

=

Lan xn converges for x e(-a-1,a-1). Then

(1) {a"

h.H

",(lR) if and only if xA (x )eM(a) (ij) {a" +1}EH ",(lR) if and only if A (x )E M (a)

(iii) {a" IEH 0. (lR +) if and only if xA (x ) . A (x )e M (0:)

(iv) {a,,+1}eH ",elR +) if and only if A (x) ,x-leA ex)-A (O»eM(a).

In view of lemma 2.3 we have the following immediate corollary to lemma 2.8:

Corollary 2.9. Let {a_n(!)l } and {an(~)l} eH (R). For i =1,2 , let Ai (x) = L an(;) xn and let

B (x)

=

Lb

n xn be formally defined by

B (x)

=

(A

leX

)]'>'1 [A 2CX

»)'l'2 ,

withi'l~ O'i'2~ Oandi'1+'Y2~ 1. Then (bn+1}eH(lR).

Using lemmas 2.8 and 2.9 Horn proved

Theorem 2.10 [6]. Let {un} and lin } be related by the renewal equations (1.1). Then (1) (un+l}EH(lR) if and only if IIn}EH(lR)

(ii) (u" }EH

elR

+) if and only if

tin

leH

(lR +)

(iii) (Un }eH([O.1]) if and only if lin lEH([O,1]) and Lfn ~ 1.

We note here that the g.L of (u"

I

(c.L(1.1» is in MCa) if and only if the g.f. of Un} is in MCa) (see [6]). But P(x) is in MCa) if and only if R'Cx) is in M'(o:) (c.f.(2.1). This is due to the fact (c.f.(2.l)) that the complex logarithm maps the upper half-plane analytically onto the strip. {z E(l I

°

<

lm(z )

<

7TL It is therefore necessary to prove two lemmas similar to lemmas 2.7 and 2.8. establishing necessary and sufficient conditions for a g.f. to belong to the set M' (a). The two lemmas will follow from

5

-Lemma 2.11. Let the function

f

be defined by

o

f(z)=

J

_z_d /-LCt).

- 0 l-zt

Im(.:; )

>

0 .

where p. is a finite measure on [-0- .0-]. Then the following equivalence holds:

/-L bounded by Lebesgue measure if and only if 0

<

ImCf(z ))

<

11'. for all z Ea with

lm(z)

>

O.

Proof: (:::;.) Clearly Im{f (.:; )

>

0 for ImCz)

>

O. Suppose /-L is bounded by Lebesgue measure. Then for Ime.:; )

>

O.

o

Im{f C.:; )):s:; 1m

J

_z

-

dt

- 0 l-zt

= Arg(1+zo-)- Arg(l-za)

<

11'.

(<=) /-L must be continuous. for if /-L had a discontinuity at t 0 ~ 0 with saltus

8.

then

let-ting =0

=

t

0

1 +i)'o. with )'0<8(t ~ 11')-1,

-Im(f (zo)) ~ _{ImCSz o/l-z ot o)}

>

11',

a contradiction. The case t 0= 0 is trivial. Observe that

o 20

-f(-(z+a)-I)=

J

1 d/-L(r)=

J

_1-

a

/-L(t-a).

- 0 z +0-+[ 0 z +t

Hence

-f

(-(Z+o-)-l) is a Stieltjes transform. Let g(x)= [ ( x + t ) -l al./l(t).

1./1

con-tinuous. By the complex inversion formula for Stieltjes transforms (Widder [14]).

lim - 21 . j[gC-cr-iT/)-gC-cr+iT/)]dcr=

1./1(0-1./1(0).

(2.2)

1) .... 0+ 11'l 0

Lets=-cr+o-+iT/. By (2.2)

lim _1_.

j

[j (-S 1)_ f (-51)] d cr

=

/-L (' - 0-)- /-L ( - a).

'I) .... 0+ 211'~ U

Thus. for any '1'~2E R . with ~1~ ~2~ -0-.

~I+o

lim _1_.

J

[j (-s-I)-

f (-

51)] d cr

=

p.(' 1)-/-LC'2)'

'I) .... 0+ 211'~ S2+ o

Hence, since ImC - z -1) > 0 if and only if Im(z ) > O. for any E >0. there is an T/ > 0 such that il+O p.C'I)-/-LC'2)-E

<

21.

J

[j(-s-I)-f(-51)]dcr 11'l ~2+o ~)+" =

~

J

Im(j(-s-I»dcr

<

'1-~2'

11' '2+0

6

We have the following two corollaries to lemma 2.11. The first corollary gives a simple generalization of lemma 2.11, the second shows that if a Moment Sequence belongs to

H' (IR) or H' (IR +), then infact all representing measures of the Moment Sequence are bounded by Lebesgue measure.

Corollary 2.12. Let the function

f

be defined by

o

fez )

=

J

- 1 z d J.1. (t ) ,

_ 0 -zt

Im(z)

>

0,

where J.1. is a finite measure on [-a ,a]. Let I E R +. then the following equivalence holds:

J.1. bounded by I·A. with A Lebesgue measure if and only if 0

<

lm(f(z)

<

[.11', for all z E(l with lm(z )

>

O.

By letting 1 tend to infinity in corollary 2.12 we regain the result of lemma 2.7.

Corollary 2.13. If a representing measure of a Moment Sequence is bounded by Lebesgue measure, then every representing measure of the Moment Sequence is bounded by Lebes-gue measure.

Proof: Let J.1. be a representing measure of {an}, and suppose p. is bounded by Lebesgue measure and has infinite support. Let J.1.i be any other representing measure of {an}, also with infinite support. Now for Im(z)

>

0

00 00 lm(

L

an zn+l)

=

ImC

J

n=O -co 00 - 1z dp.(t»)~ ImC

J

_Z-dt)=7T. -zt - 0 0 l-zt Hence for lm(z )

>

0, 00 0

1T~Im(

J

-z-dJ.1.,(t»>Im(

J

- 1z dJ.1.i(t))>O.

_ex> l-z( - 0 -zt

Since a is arbitrary. but finite. we see, appon applying lemma 2.11. that P.i must also be bounded by Lebesgue measure .•

It is not possible to strengthen this statement by asserting that J.1. is unique. Feller ( pg. 224 [5]) gives an example of a set of measures on [0,00) having the same moments and each measure is bounded by Lebesgue measure. We are now ready to prove the N' (a)

analogues of lemmas 2.7 and 2.8.

Lemma 2.14. Let O'ER + and let A be a real valued function defined on (-a ,a). The fol-lowing are equivalent:

0) A(x)eN'(a)

(ji) A(x)= Lan xn with xe(-a-1,a- 1) and (an+l}eH~(R)

(iii) there exists a finite measure p. on [-0'-1,0'-1] bounded by Lebesgue measure such that

"

7

-Proof: In view of lemma 2.7, we need only prove the equivaJence between the following three statements:

(n

A(x) maps the upper half-plane into

1=

Ea; 10< Im(z) < 1T}.

(in

the representing measure of

la

li +

d

is bounded by Lebesgue measure.

(iii') the measure p. defining A(x) through (2.3) is bounded by Lebesgue measure. By lemma 2.11

(n

and Or) are equivalent. By taking the g.f. of {a" } in

(in

we obtain Wi'). On the other hand, by (2.3)

00 ()

A

ex )

=

00+

.E (

J

t n d p. (t )) x n + 1 •

n::::O -()

thus by the uniqueness theorem for g.f.s we must have {an +l}eH: (lR ) .•

Lemma 2.15. Let a e I( + and let {a,,} be a sequence of real numbers such that

A(x)= .Eon xn convergesforxe(-a-1,a-l).Then

(0 (a" IEH: (lR) if arul only if xA (x )EA1' (a) (ii) {an +tlEH~ (lR) if arul only if A (x )E J'yl' (a)

(iii) {an }eH: (lR +) if arul only if xA (x )e M' (a) and A (x )e M (a)

(iv) {an +lleH: (lR +) if arul only if A (x )e M' (a) and x-leA ex) - A

(O»e

M (a).

Proof: Part (0 and Part (ij) follow directly from lemma 2.14. We now give the proof of Part (iii).

(:::;. ) Follows from Part (j) and the fact that

H:

(I< +)C

H:

(lR) together with Part (iii) of lemma 2.8 and the fact that H: (lR+)CH ()(lR+).

(<= ) By Part (i) above we have that ian }eH: (lR) and By Part (ij) of lemma 2.8 that

Ian +l}eH ,,(lR). Hence by lemma 2.8 Part (iii) together with the fact that

H: (lR)C H () (.R), we have {an }eH 0 (lR +). Let p.' be the representing measure of

Ian IEH: (lR), and p. a representing measure of {an leH ,,(1(+). Then

Iml

j

-Z-dp.*(t)

=Im

j

Z dp.(t) . _" 1-zt 0 1-zt

(2.4)

By lemma 2.14 the left hand side of (2.4) is positive and bounded above by 17', thus the

right hand side is also positive and bounded by 17'. so by lemma 2.14 p. must be bounded

by Lebesgue measure. Hence {an} e

H:

(I( +). A similar argument holds for Ov).'

In the following. if we write that both {an} and {b_{n }} are members of H e>:(lR),

H: (lR). H 0(1<+) or H: (1<+), it will be understood, for ease in notation. that the a's

corresponding to {an} and Ibn } need not be the same.

Remark. In proving the (1.2) equivalence of theorem 2.10, we will only prove the result for moment sequences whose representing measures have finite support. Once the result is established for measures with finite support, the following argument can be applied to obtain the result in full generality. If (P,. ) is a moment sequence. then there exists a sequence of moment sequences. {Pn (T») say. whose representing measures have finite sup-port and such that Pn (T ) - Pn . Define {rn (T)} using (p" (T)} in (1.2). Then, as we will see, {rll (T)/(n +])) is a moment sequence for every T, with its representing measure

hav-ing finite support and bounded by Lebesgue measure. The limit of (rn (T»); {rll ) say. is

8

-{r" I(n

+

1)) is therefore a moment sequence with its representing measure bounded by Lebesgue measure. A similar argument holds when starting with

ir" },

Thus the theorems will be stated in full generality. but the proofs will only be carried out for measures with finite support.

3. Hamburger Moment Sequences

In this section we will prove two theorems concerning the Hamburger Moment behavior of two sequences related by (1.2). i.e by

n

(n +t)Pn+l

=

L

rk Pn-I. .

t=o Po

=

1 '

The first theorem states a relationship between (Pn +1) and frn I(n

+

I)}. the second theorem between {Pn) and {rn I(n +t)}. As a corollary to the second theorem we obtain a relation-ship between {Pn } and frn +1/(n +2)}.

Theorem 3.1.

Proof: From lemma 2.8 we have that {Pn+l}EH

0'C10

if and only if P(x )EM(a). Since: R' (z )

=

log(P(z)) .

by the properties of the complex logarithm. R' is constant if and only if P is. and R' maps the upper half-plane into the strip. {z eq; I 0

<

Im(z)

<

TTL if and only if P maps the upper half-plane into it self. Hence P(x h.:M(a) if and only if R' (x )eM' (a). By lemma 2.15 (ij) it follows that R'

ex

)e M' (a) if and only if {rn I(n

+

J)}€H~ (JR ) .•

The proofs of the following theorem and its preperatory lemma are very similar to the proof of theorem 2.12.1 in [11].

Lemma 3.2. If t J. . . ,tN and SI,' .. ,SN-I are given. satisfying

tAo

<

S"'-1

<

(~'-l

< ... <

S1

<

t 1 •

then there exists unique q 1

>

0, ... ,qs

>

0 such that

'"

L

qk t:=1 1- xtk

has zeros S

11 , ...

,S"'':1 and Lt~ 1 ql.

=

1.

Proof: Expanding II;'\~"'il(1-XSj )/II/~ll(l-xt,) into partial fractions. we see that

(3.2)

s I N - I S ql.

II

II

(l-xSj)

=

L

(3.3)

i=1 1- xtj )=1 k = l l - x t t

where

The inequalities (3.2) imply that ql.

>

0 and putting x

=

0 in (3.3) shows that Lqk

=

1.

The uniqueness follows from the uniqueness of both the partial fraction expansion and the factorization of polynomials .•

9

-We are now ready to prove a theorem which determines the relationship between the moment behavior of {Pn } and Irn I(n

+

1)} (compare with theorem 3.1 and lemma 2.5). Theorem 3.3. Ip" IEH(JR) if and only if there exists two sequences Ibn

Lien

}EH'(JR+).

such that

rn I(n +1) = bll - (-IY en

Proof: (:::;;.) Suppose the representing measure is discrete with atoms qj at

ti • (i

=

1.2 ... N). where -0' <tp; < ... <tl+1 <O<t, < ... <t] <0'. The g.f. of {p" } is then P(x)=

t

q/.. /..:::: 1 l - xtk l\'

=

Q(x)

IT_I-t=1 1- xt"

with Q a polynomial of degree at most N-J. Observe that

P(x)

<

0 for x !t/..-l • for k

=

1.2 ... 1

P(x)

>

0 for x 1t/..-1 • for k = 1.2 ....

.1

P(x»Oforxitk- 1 .fork =l+1. .... N P (x )

<

0 for x tt A I . for k = I

+

1 ... N

thus. since Q is continuous on R except at tl.:-1• k

=

1.2 .... N. we know that N-2 of the N-J zeros of Q. denoted by s,,-I. k

=

1.2 .... .l-1,1 +2 .... ,N , are located as follows:

-O'<tN <sN < ... <SI+2<t'+1<0<t[ <s/_I< .. ·<SI<tl<O"

If a zero of Q, s-1 say (s ;:C 0). is the site of a local extremum of Q we must have: a

0= P'(x) 1,,=,-1=

J (

t_l )2 dp.(t)

-a 1-s I

0= P(x ) I .\

which implies that

Le. that J.I. = O. a.e . . Thus no zero of Q is the site of a local extremum: hence the reciprocal of the N-J st zero. s say. must lie in (tl+l,t,). Let

S'+1 = minCO.s)

Sl

=

max(O.s ).

If Q only has N-2 zeros (this corresponds to s = 0 above). then let s, = .)/+1 = O. In either

case:

}; l\'

P(x)

=

IT

Cl-x.)/.. )1

IT

(l-xt/.. ).

/..::: 1 ,(:::: 1

with Hence 10 -N R' (x) = 10gP(x) =

r.

[-log(l-xt~ )]- [-logCl-xs.l.)] m·.r.,·l(t )

=

t

=

1 I lk ". Sk

J

X .

J

x

=

r.

- - d t -

r.

- - d l .l.=ls k I-xl .. =1+11k I-Xl ~ ~ = J _x_ dmN l(t ) - J _x_ dmN2(t ). o l-xt 0 l+xt

I

0 1 if t E (s ... t t ) • otherwise

1

0 1 iftE(-S.l..-t .. ) . otherwise k

=

1.2 •...• Z k

=

Z+1 ... N o 0 rnl(n+l)= J t n dm .... ·1(t)-(-l)n Jt n dm"'·2(t). o 0

Let the representing measure of the sequence {Pn

I

be IL. By the remark at the end of section 2. IL may be assumed to have finite support. It is easily verified that P can be obtained as

N

P(x) = lim P,.,·(x) = lim

r.

qu .. '

N-= N-=t =1 l-xt .. .N

= lim expl

j

_x-dm""l(t)-

j

-x-dmNit

)1.

N-= 0 l-xt 0 1+xt

By Helly's first theorem there exists subsequences {mN_k

dt'=

1 and {mNI21t"~ 1 converging

weakly to the measures mI. m2 resp. Thus by Helly's second theorem we have. for fixed x

~ 0 lim J _x_ dmr.,· l(t)

=

J _x_ dm l(t ) .. - = 0 1-xt . k 0 1-xl o 0 lim J _x_ dm\. i t )

=

J _ x - dm2(t) . I - 00 0 1

+

xt . I 0 1

+

xl

By lemma 2.4 all convergent subsequences of {m N11 must converge to ml and all

conver-gent subsequences of {mN21 _{must converge to m2' Hence {mN1}1 converges weakly to m1 and {m.nl converges weakly to m2' Hence

P(x)= exrl

j

_ x - dm1(t)-

j

_{_ x - dmit}

)1.

o I-xl 0 l+xt

with ml and m2 bounded by Lebesgue measure. Thus

o 0

rn I(n +1)

=

J t n dm l(t) - (_l)n J t n dm2(t).

o 0

(3.4)

For k = 1.2 .... ,N -1 , let

.

_11

iftE(Sk,J\·,t.,A')

m A'l(l ) - 0 otherwise

mA' l(O.cU\']

=

m l(O,t. ,N] ,

For k

=

N .N

+

1.," .2N -1 . let N-k N

.

11

if t

d-s

k )\, , - tk ,11') m A'2(t )

=

°

otherwise ml\'2(0.-SJ.~"':]= miO,-st,]I'] ,

Then we have for any t E(O.o:]: ~im mlO(O,t]

=

mICO.£]

11-00

lim m.\'2(O.t]

=

miO,t],

}.' - 00

J J

-thus. using Helly's second theorem and lemma 3,2. we have

P(x)= exrl

j

_ X -dm1(t)-

j

- 1 x dmh)1

o I-xl 0 +xt

=

lim expl

j

_x-dmx1Ct)-

j

_ X -dmN2(t)1

J,' - 00 0 1-Xl 0 I

+

xl .

=

lim exp

Atl

J

x dt _

21: 1 ]

_x_ dl

1\'-00 1:=1 SA I-xl k=;A.,f tit 1-xt

2:\,-11-X51.

=

lim

IT

12 -21\'-1

=

lim

r.

q~ II: - 00 /.: '" 1 1-xl k o = lim

f

_1_ d J.l y (1 ) . lV-co _ 0 I-xl .

By Helly's first theorem we have a convergent subsequence, {J.l_Nk

lk':l

say. converging weakly to some measure J.l. By Helly's second theorem

0. I 0 1

lim

f - -

d J.l y (t )

=

f - -

d J.l(t).

I.: - 0 0 - 0 1-xl . i< - 0 l-xt

By lemma 2.4, all convergent subsequences of {J.l.l\' } converge to J.l, hence

o

P(x )

=

_I

l~xt

d J.l(t ) .•

Corollary 3.4. {Pn}eH(lR) if and only if {rn+l!Cn+2)}eH(lR). with representing measure J.l satisfying:

- 0 0

(ij) [d J.l(t )

~

[I

t

I

dt , B a Borel set.

Proof: We need onl y prove

tr

n I(n

+

1)} is as in the statement of theorem 3.3 if and only if {rn+t1(n +2)}eH (lR) with representing measure satisfying

en

and (ii).

C=»

Let m 1 and m2 be the representing measures of {an }. {b_{n } resp. in theorem 3.3. Let}

d J.ll

=

t dm 1 and d J.l2

=

t dm2' Also let

j

J.lz[O,OO) - J.lz[O,-rJ • t

~

0

J.l(-oot]=

, J.l2[0,oo)+J.ll[0,t) .t>O.

Then {r" +d(n +2») EH (lR) with its representing measure satisfying (i) and (ii).

(3.5)

(<=:) By (j)

L

t-1d J.l(t) or

L:

t-1d(-J.l(-t» is non-negative and finite. Assume (after rescaling if necessary).

co

f

t-1dJ.l(t)= ro~ O.

- 0 0

Let dml=t-1dJ.ll and dmz=t-1dJ.lz. where J.ll and fJ.2 are defined via (3.5). Then tr" I{n

+

I)} is of thI:0rm (3.4) with m 1 and m2 bounded by Lebesgue measure. A similar

argument holds if t-ld(-J.l(-t»~ 0 . •

- 0 0

4. Stieltjes Moment Sequences

In this section we prove the analogue of theorem 3.1 for StieItjes Moment Sequences. Theorem 4.1.

-13

Proof: (::::;. ) By lemma 2.1 IPn } , (Pn + II E H 0 (IR). This implies. in view of theorem 3.1 and

corollary 3.4, that

Irn

I(n +1)}EH~ (R) and {r" +l/(n +2)}EH o(IR). Hence by lemma 2.14

Part (i), R'(x)EM'(O'.) and by lemma 2.8 Part (ii) x-1(R'(x)-R'(x»EM(0'.). Thus

(r" I(n +1)IEH~ (lR+) by lemma 2.15 Ov).

(<=) By theorem 3.3 (with b"

=

0) we have {p,,}EHc.(IR). Since H~(IR+)Cll~(IR).

then by theorem 3.1 we also have that {Pn+l}ElI o(IR). Hence by lemma 2.1,

(p" IEH oCR +).1

We have the following immidiate corollary:

Corollary 4.2. Let {ant!)} and {a,,(2)}EH(IR+). For i=1.2 ,let Ai(x)= L,a,,(i)x" and let

B(x)

=

L,bn Xli be formally defined by

E (x)

=

[A leX )fl [A 2(X )]'>'2.

with)'l ~ 0')'2 ~ 0 and)'1 +)'2 ~ 1. Then {bn )EH(lR+).

We now give a proof of theorem 4.1 which does not use the result of theorem 3.3 Proof: (of Theorem 4.1) In terms of g.f.s. we must prove (c.f. lemmas 2.8 (iii) and 2.14 Ov»

From theorem 3.1 we have

P(x)EM(O'.) <:;> R'(X)EM'(O'.).

We need therefore only prove

(j) P(x) and xP(X)E MCO'.)::::;, x-IR' (x )eM(O'.)

(ij) R' (x ) EM' (0'. ) and x -1 R' (x )E M (0'.) ::::;. xp(x )e M (0'.).

(0.

Let the g.f. of the sequence {s" }. denoted by S, be defined by 1

P(x)= 1-xS(x)

i.e. {p"

I

and

Is"

I

are related by the renewal equations (1.1), Thus by theorem 2.10 (j)

(Pn}eH oCR +) if and only if

Is"

JeH

",(lR+), Now

Im[z-l_{R' (z)] = 1m [-z-}1_{10g(l-zS(z »]} 1 = 1m z-1

J

zS(z) dt o l-zS(..:)[ 1 1 1

=

ImCS (z»

J

( )

2 dt

+

Im(z)

J

o 11-zS

z

I 0

since lm(z)

>

0 and {s" IEH oUR +), which in turn implies

{>

Im(S (z

»

=

1m

J

_1_ d p.(t )

o l-zt

=Im

J

o

-zt

2 d P. (t)

~

O.

14

-So z-IR' is either constant Cif P =: exp(z) ) or maps the upper half-plane into itself. Since

the complex logarithm and P are both analytic. then z-IR' is also analytic. Thus

x -1[R' (x )-R' (O)]e M (0').

(ii). Since {rl1 I(n + 1)} E H: (R +), we can write R' as

w 0

R'(z)= Lr/l!Cn+l)z"+1=:

J

_z-dJ.LCt).

n 0 0 1-zt

Thus, letting Im[R' (z)]

=

1 (z ) and Re[Rs (z)]

=

J (z ) and z

=

x + iy; Im[R' (z )] =: 1m

I

j

_z_

d J.L(t )

I

~

lm[-log(l-za)]

(l 1-zt

=: coCI a-1-x

I

~

coCI _-x

Y

y

hence

y cos(J(z » + x sinU (z

»

>

O. But

Im(zp (z ))

=

[y cosU (z

»

+

x sinU (z

»)]

explJ (x

»).

Since xP( x) is real valued and regular in (-0' -1,0' -]). then xP (x )e M (0' ) .•

5.

Hausdorff Moment Sequences

By using induction and the fact that {rn ) and {Pn } are related by (1.2). one can prove

that {Pn} is bounded if and only if {rn I(n +l)} is bounded. In view of lemma 2.2. it fol-lows from theorem 4.1 that

Theorem 5.1. {Pn }EH([O,l]) if and amy if {rnl(n+l)}EH'([O,l]).

Corollary 5.2. Let _{{a n}(1)} and {an(2)}EH ([0.1]). For i =1,2 , let Ai (x) =

1:

anU) xn and let

B (x) =: 1:b" x n be formally defined by

B (x)

=

[A

leX »"1

[A 2(X

)]Y2.

WithYl~ 0'Y2~ OandYI+Y2~ 1. Then !bll}eH([O,l)). 6. Applications to Probability Theory and Renewal Theory

In this section we discuss two applications of the results of the previous sections. First an application is found in the theory of infinitely divisible lattice distributions on

1l +. more precisely to those which are completely monotone. Second we find an applica-tion in the field of Harmonic Renewal Sequences.

Completely Monotone Lattice Distributions

A sequence {a" } is said to be Compl£tely Monotone if for all ~ Ell + and all r E Z +.

r

1:

<[) (-lY: ak+~ ~ O. k (\

It is wel1 known (c.f.[9]), that a sequence is completely monotone if and only if it is a Hausdorff Moment Sequence. Steutel [10) proved that any completely monotone

- 15

probability distribution is infinitely divisible. Theorem 1.1 can rewritten as follows (c.f.[ 12])

Lemma 6.1 [12]. A probability generating function P. with P(O)

>

0 is infinitely divisi-ble if and only if P can be written as

P(xl=

exrl-,J

R(Ulduj

where R is a generating function with positive coefficients and necessarily

1

J

R(u )du

=

log(p(O)-l)

<

00.

()

The following representation theorem is the exact analogue of theorem 2.12.1 in [11] for completely monotone lattice distributions on iZ +:

Theorem 6.2. A probability generating function P is of the form

1

P(x )

=

J

I1- t dF(t)

o - x l

with F a distribution function on [0.1] if and only if P can be represented as

where

P(x

1

=

expl-,J

R(u ldu

j

1

R (u )

=

J

(l-uv )-2 dm (v ) o

and m is bounded by Lebesgue measure. Both F and R are unique.

(6.1)

Proof: A g.f. is of the form (6.1) if and only if it is the g.f. of Sequence with PU)

=

1 (c.f.[lO]). Hence by theorem 5.1

a

Hausdorff Moment

I

I

I

P(x) x

- - =

exp

,J

- - d m ( v )

P(O) (. I-xv (6.2)

with m bounded by Lebesgue measure. But

I 1 1

J

1: .

dm (v)

+

log(P(O))

=

J J (l-uv )-2 dm (v )du

o

x.

0 0

I 1

- J J(I-uv )-2dm(v )du

+

logCP(O»)

x 0 1 1

= -

J J (I-ut' )-2dm(v )du.

\ 0

16

Thus far we have been able to determine the relation between some integral represen-tations of two g.f.s related by (2.1). But no direct relationship between the representing

measures of the two g.f.s has been established. If we let (U = -log(P(O)) and

(UdF I(V)

=

(1-v )-ldm (v) in (6.2), then (6.2) can be rewritten as;

P(x)

=

exp!-(U

+

(ux

JI

I-v dF I(V)

I

o I-xv

where F 1 is a distribution function. So the special case where F

=

F 1 a,e .. (c.f.(6.l)).

gives rise to the expression

P(x)= exp{-(U[l-xP(x)]}. (6.3)

Here we have in fact assumed a direct relationship between the representing measures of P

and R'. We note here that a discrete probability distribution {Pn} is called a generalized poisson distribution with papameters

«U

,e) if

(U

«U

+

n

e

)n -1

Pn

=

,

exp{-w-n6}, n

=

0,].2... (6.4)

n.

We have the following lemma concerning (6.3). Lemma 6.3. A probability g.f. P is of the form

P(x ):: exp{-w[l-xP(x )]} . (6.5)

with w

=

-log(P(O)). if and only if P is the g.f. of a generalized poisson distribution with

parameters (w.w).

Proof: (::::;.) If a P exists satisfying (6.5). then P is infinitely divisible. Hence by (2.1) we must have

R' (x ) = (Uxp(x ) .

i.e.

rn :: (n

+

1) w Pn . (6.6)

By using (6.6) in (1.2), we can solve. using induction. for Pn in terms of Po. to get that

{Pn

I

is of the form (6.4) with w :: 6

=

-Iog(P(O)).

(<= ) Let Pn be given by (6.4) with w

=

6

=

-log(P(O)). Define {rn} through (6.6). It can be shown that (rn } satiefies (1.2) with (Pn

J.

Then

P(x ):: P(O) exp{R' ex )}

=

exp!-(U[l-xP(x )]} .

Hence a solution exists and is given by (6.4) with w

=

e

=

-log(PO))

,I

It still remains to be shown that the generalized poisson distribution with parameters

(w,w) in fact is completely monotone. To prove this we need the following lemma, proven

by Bouwkamp [1].

Lemma 6.4 [t]. The fo11ov..'lng equality holds; sinx - - exp(x cotx) x n 1Tnn dv _{: :} -A n. " (6.7)

17

-Lemma 6.5. The generalized poisson distribution with parameters (CLI ,CLI) is completely

monotone.

Proof: Let A(x} denote the function in brackets in (6.7). A'(x} is negative on (O:IT] and so its inverse exists. Call it G(y). Then

TTn n "IT

- - =

1

[A (x )]11 dx n! 0 e

= -

1

[y ]n G' (y ) dy o e

=

1

nun- 1G(u)du . o (n

+

1)"

-we

-w)n_ (n

+

1)! e CLI e -we-w +1

1

VlldI-LI(V). o

Observe that CLI e-w + 1 ~ 1 for CLI ~ 0; letting the measure I-L be defined by

!

I-LI'(V)

ifv~CLle-w+l

I-L ,( v)

=

0 otherwise we have Pn

=

(n

+

1 )n

-we

-w)n _

11

n d ( ) (n

+

l)! e CLI e - 0 v I-L v ,

where I-L is bounded .•

Harmonic Renewal Sequences

Let the g.f. of the sequence {gn} be denoted by G, and let {In}, whose g.f.will be

denoted by F. be a probability distribution on Z +. If G is of the form

00

G(x)= Lbm[xF(x)]m,

m=O

then {gn

I

is called a Generalized Renewal Sequence (c.f.[3]). If bm

=

1. m E Z +. then {gn

I

reduces to the the renewal sequence corresponding to {tn}. If b 0

=

O. bm

=

m -1 for m ~ 1

we call {gn

I

the Harmonic Renewal Sequence corresponding to {In}. Corresponding to

Un)

let {un

I

be the renewal sequence and {rn

I

the harmonic renewal sequence. Let F. V, and R denote the g.f.s of {tn },lu_{n } and {T'n} resp. We then have the following relationship} between V and F;

V(x)

=

[l-xF(x )]-1 .

The relationship between F and

R

is given by;

l-xF(x)= exp{-R(x )}.

(6.8)

(6.9)

If we use {un} in place of {Pn} in (1.2). and assume that V and F are related by (6.8).

then {r" I(n

+

n} in (1.2) corresponds to {rn}. From theorems 2.10. 3.1. 4.1 and 5.1 we

- 18

Theorem 6.6. Let the g.f.s of (f"

l.

IU

n

I

and

IP" I

formally be related by (6.8) and (6.9).

The following three statements hold;

0) (f" IE H ([0,1]) if and only if

lUll

IE H ([0.1]) if and only if

lin

IE H' ([0.1]).

(ii)

lin

}EH (lR+)

if

and only if

lUll

hH (lR +)

if

and only

if

IP

n

IEH' (lR +).

19

-References

[1] Bouwkamp C.J .. will appear in SIAM Review. as solution to problem 85-16. SIAM Review. 27 (1985). pg. 573.

[2] Bruijn N.G. de. Erdos P .. Some Linem' and some Quad7-atic Recursion Formulas. lndag. Math .. 13 (1951). pg. 374-382.

[3] Embrechts P .. Omey E .. Functions of Power Series. Yokohama Math. J .. 32 (1984). pg. 77-88.

[4] Erdos P .. Feller W .. Pollard H .. A Property of Power Series with Positive

Coefficients. Bull. Amer. Math Soc .. 55 (1949). pg. 201-204.

[5] Feller W .. An Introduction to PrdJability Theory and Its Applications, Vol. II. Wiley. New York. 1966.

[6] Horn R.A .. On Moment Sequences and Renewal Sequences. J. Math. Anal. Appl.. 31 (1970). pg. 130-135.

[7] Kaluza T .. Uber die Koefficienten Reziproker Potenzreihen . Math. Z., 28 (1928). pg. 161-170.

[8] Katti S.K., Infinitely Divisibility of Integer Valued Random Variables. Ann. Math. Stat.. 38 (1967), pg. 1306-1308.

[9] Shohat 1.A.. Tamarkin J.D .. The PrdJlem of Moments. Surveys I. Amer. Math. Soc .. Povidence R.I.. 1943.

[10] Steutel F.W .. Note on Completely Monotone Densities, Ann. Math. Stat.. 40 (1969). pg. 1130-1131.

[11] Steutel F.W .. Preservation of Infinite Divisibility under Mixing and Related

Topics. Math. Center Tracts 33. Math. Center. Amsterdam. 1971.

[12] Steutel F.W .. Harn K. van. Discrete Analogues of Self-decomposability and

Stabil-ity. Annals Prob .. 7 (1979). pg. 893-899.

[13] Warde W.D .. Katli S.K .. Infinite Divisibility of Discrete Distributions 11, Ann. Math. Stat.. 42 (1971). pg. 1088-1090.