Degrees of infinite divisibility, and a relation with p-functions

Degrees of infinite divisibility, and a relation with p-functions

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Harn, van, K. (1977). Degrees of infinite divisibility, and a relation with p-functions. (Memorandum COSOR; Vol. 7701). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1977

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PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 77-01 Degrees of infinite divisibility,

and a relation with p-functions by

K. van Harn

Einhoven, January 1977 The Netherlands

Dearees of infinite divisibility, and a relation with p~functions

Sunnnary

In [2J a classification of the infinitely divisible (inf div) lattice dis-tributions was given by means of recurrence relations. In this paper this classification is extended to all inf div distributions on [0,(0); we intro-duce a system of increasing classes FA (0 ~ A < (0) of inf div distribution functions F on [0,(0), where F E FA iff F satisfies the functional equation

x x

J

(J - e -AY )/(1 - e )dF(y) -A =

J

F(x-y)dKA(y)

a

a

for a KA that is nondecreasing.

Fa

is known (see [7J) to be the set of all inf div distributions on [a,""), while F"" ,- lim FA turns out to be a set of

A -l>«>

distributions, containing all log-convex, and so all completely monotone, densities (possibly with a jump at zero), and which also contains (save for norming) both the renewal sequences and the standard p-functions of Kingman.

Degrees of infinite divisibility, and a relation with p-functions

by

K. van Harn

Eindhoven University of Technology, Eindhoven, The Netherlands

1. Introduction

. 00

The class

r

₁ of infinitely divisible (inf div) lattice distributions {Pn}o with Po >

°

(i.e. the class of compound-Poisson lattice distributions) can be characterized by (see [3J or [8J): {p } ~

r

1 iff the quantities r (1),

n n for n = 0,1,2, ••• defined by (1. 1) (n +1)p 1 = n+ n

L

Pkr -k(l) k=O n (n = 0,1,2, ••• ) ,

are all nonnegative. The class rO of compound-geometric lattice distributions can be characterized similarly (see [7J): {P

n} € rO iff the quantities rn(O), for n'" 0,1,2, ••• defined by

(1.2) (n = 0,1,2, ••• ) ,

are all nonnegative. In [2J these recursion relations were generalized to obtain classes

r

of inf div lattice distributions (0 ~ a ~ 1) with the

pro-a

perty

(1.3)

The

r

are defined as follows. a

Definition 1.1. A lattice distribution {Pn}~ with Po > 0 is in the class ra if the quantities r (a), for n

=

0,1,2, ••• defined by

n (1.4) n+l - a 1 - a n Pn+l

=

k~O

Pkrn-k(a) are all nonnegative.

(n ... 0,1,2, ••• ) ,

Remark. From Definition 1.1 we obtain classes of generalized renewal sequen-ces {un}~' if we replace the condition that ~~Pn

=

1, by the requirement that U

_{o ::}

1. Especially, taking a ... 0, most properties of distributions in

2

-ro can be translated into p-roperties of renewal sequences, and vice versa. A similar correspondence exists between a class F' of inf div densities (to

«>

be defined later) and the class of standard p-functions (cf. [4J, Theorem 3.1 and our Theorem 5.9).

In this paper we give a similar classification of al1 inf div distributions on [0,«». The starting point for this is the absolutely continuous analogue of (1.1) (see [7J): a probability density function (pdf) f on (0,«» is inf div iff there is a nondecreasing function K such that almost everywhere

( 1.5) xf(x)

=

JXf(X -

y)dK(y) .

0-For pdf's there is no obvious analogue of (1.2); this would be

(1. 6) f(x) =

(f(X -

y)dK(y)

0-with a nondecreasing K. However, (1.6) is satisfied by all pdf's if K is the unit-step function and by none for any other K. To overcome this difficulty we first generalize the classes r for a > O. The set of absolutely

continu-a

ous distributions in the intersection of the resulting classes can then be considered as the analogue of rOo

Proceeding as in the discrete case we replace the factor x in the left-hand side of (1.5) by the function

(0 ~ A < «>; X ~ 0) ,

with c(x;O) := x, Considering, more generally, distribution functions (df's) rather than pdf's, we are led to the following definition of the classes FA'

Definition 1.2. For 0 ~ A < «> a df F on [O,~) is in the class FA if there is

a nondecreasing function KA onR, with KA(X)

=

0 for x < 0, such that for all x > 0 (1. 7) x

J

c(YjA)dF(y) = 0-x

f

F(x - y)dKA(y)

For A m 0 we obtain the class FO of all inf div df's on [O,~) (see [7). It

is convenient to introduce the following subclasses of FA for 0 ~ A < ~

( 1.8) Now we we get ( 1 .9)

F

+ o -X .-F' := A formally c(x;~) F(O) > O} , F absolutely continuous} • let A -+ co in (1. 7) • As

~

if x >

o ,

:= lim c(x;A) ... A+= if x ...

o ,

x

F(x) - F(O)

=

f

_{F(x - y)dK (y) •}

<X>

0-If F(O)

=

0, the same difficulty arises as in (1.6) for pdf's, but if F(O) >0, (1.9) makes sense; the df's F with F(O) > 0 and satisfying (1.9) with a non-decreasing K~ correspond to the compound-geometric distributions on [O,~)

(see

[7J),

which are known to be inf div. This class of distributions we de-note by F+ (cf. _co (1.8», as it turns out to be the subset of df's F with F(O) > 0 in a class F to be defined later. So we have the followipg defini-_co tion.

Definition 1.3. A df F on [O,~) is in the class F+ if F(O) > 0 and if there

00

is a nondecreasing function K onE, with K (x) _{co co}

=

0 for x < 0, such that (l.9) holds.

In section 2 we prove a number of preparatory results, some of which may be of some independent interest. In section 3 we prove the monotonicity of the FA (cf. (1.3» and consider the absolute continuity of F in FA in terms of the function K

A, Here we also introduce the class Fco:= _A

n

FA containing the

<00

continuous analogue of rOo In section 4 properties of the classes FA are dis-cussed, and examples are given. In section 5 the classF , which (save for ₀₀ the proper norming) contains both the renewal sequences and the standard p-functions of Kingman (1972), is considered separately, and briefly.

For general properties of inf div distributions we refer to [IJ, chapter XIII and [5J, chapter 5.

4

-2. Preliminary results

.We first introduce some notation. The Laplace-Stieltjes-Transform (LST) of a nondecreasing function U on [o,~) is denoted by

6(1'):=

J

e -TX dU(x).

0-The corresponding small letter u will be used for the density of U in case of absolute continuity. The ordinary Laplace-Transform (LT) of u is then

A A

also denoted by U(.). I f U is a df on [o,~), we shall say that U .is a pro-bability-Laplace-Stieltjes-Transform (PLST). Finally, if a df F is in the

...

class FA' we shall also say that F E FA' and, in case of absolute

continui-ty, that f E F~.

We first consider the class F 0 in more detail. For ease of reference we state the well-known representation theorem given in [IJ.

Theorem 2.1. A df F is in FO iff the function

is completely monotone (comp mon), or, equivalently, there is a nondecreas-ing function KO on~, with KO(x) == 0 for x < 0, and

~

(2. 1)

f

x dKO(x) -I < ~ ,

1 ....

such that F has the form

(2.2) F(.) == exp[

f

0--1

l)x dKO(X)]

The function K

O' which is uniquely determined by F, will be called the cano-nical function of F. We shall need the following properties.

Lemma 2.2. Let F E FO with canonical function KO' Then we have

i) KO(O)

=

inf{x > 0 F(x) > O}, . ii) If KO(O) == 0, then

(2.3)

J

x dKO(x)

-)

< ~- F(O) > 0 • 0+

Proof. Define y := inf{x > 0

I

F(x) >

oJ.

The df G(x)

:=

F(x +

y)

is again inf div, with canonical function L

O' say. According to (1.7) for all x > 0 we have x G(x)LO(O)

~

J

G(x - y)dLO(Y) ... 0-x

J

ydG(y)

o

~ xG(x) ,

and as G(x) > 0 for all x > 0, it follows that LO(O) ...

o.

But using (2.2) we then have

~ -yTA

I

-Tx-1

F(.) ... e G(-r) - exp[-y. - (l -e )x dLO(X)]

0+

from which by the uniqueness of the canonical function it follows that KO(O)

=

y. If KO(O) ... 0 we can write

-log

F(.)...

J

(1 0+

and letting. + ~ part ii) of the lemma follows.

o

Using this lemma we obtain the following r~presentation theorem for the class +

FO'

Theorem 2.3. A df F is in F~ iff F is compound-Poisson, i.e. F has the form

...

(2.4)

where ~ > 0 and G is a df with G(O) ...

o.

+

Proof. Let F E FO' As F(O) > 0, from the preceding lemma it follows that KO(O)

=

0 and so

~:-

J

X-JdKO(X) <

~

• 0+

(

- 6 - l

x

G(x) :=

~-l

f

y-IdKO(Y) , 0+

then G is a df, and it is easily seen that (2.2) can be rewritten in the form (2.4). Conversely, it is well known that a PLST F of the form (2.4) is inf div with F(O) > O.

As we noted in section 1, for F+ there is a similar representation theorem.

00

Theorem 2.4. A df F is in F+ iff F is compound-geometric,

00

....

Le. F has the form

(2.5)

where 0 ~ p < 1 and G is a df with G(O)

= o.

Finally, we state a property of FO which we need in section 3 to prove the

monotonicityof the FA'

Lemma 2.5. If F E FO and a > 0, then Fa(T) := F(a)F(T)/F(T + a) is a PLST in FO'

o

Proof. Let KO be the canonical function of F, then one easily verifies that Fa is of the form (2.2) with canonical function

K~a)

given by

Next we K(a) (x)

=

o

o

consider df's

I

x -ay (1 - e )dKO(Y). in FA for A > O. For c A := lim C(X;A) = (l -A -I - e )

.

x-+oo

Then, taking LST's in (1. 7) and (1.9), we (2.6) and (2.7)

o

o

< A ~ 00 define get

Now Bernstein's theorem immediately yields a characterization of the classes FA (0 < A < ~) and F: in terms of LST's (cf. the first part of Theorem 2.1). As it is not clear, that for an arbitrary df F on [O,~) there is a function KA of bounded variation such that (1.7), or (2.6), holds, we first introduce the following functions ~A' noting that ~A coincides with KA if F € FA'

Definition 2.6. For an arbitrary df F on [O,~) we define

i) _{~A Cr)}

ii)

;= cA{1 - FCT + A)/P(T)} for 0 < A < ~ ,

;= lim ~A(T) = -F'(T)/P(T) , HO

and, if furthermore F(O) > 0,

iii) ~~(T)

:=

lim ~A(T)

=

1 - F(O)/F(T) •

A~

Lemma 2.7.

i) For 0 ~ A < ~ a df F on [O,~) is in FA iff the function ~A is comp mono ii) A df F on [O,~) is in F+ iff F(O) > 0 and the function ~ is comp mono

~ ~

We shall need the following lemma, which does not seem to be generally known.

Lemma 2.8. If F is an arbitrary df on [O,ro) and a > 0, then

(2.8) lim F(T + a)/F(T) = exp[-aYFJ ,

T~

where YF := inf{x > 0

I

F(x) > OJ. Proof. First let Y

F

=

O. For each T > 0 there is a e(T) € (0,1) such that

Hence we can write

o

5 I - F(T + a)/F(T)

=

-aP'(, + 8(T)a)/F(T) s -apl(T + 6(T)a)/F(T + 8(T)a) ,

from which it follows that for (2.8) it is sufficient to prove that

(2.9) lim -F'(T)/F(T)

=

0 •

8

-""

A

To do so we use the identity F(,)

= ,

r

e-T~(x)dx, and obtain for € > 0

J

o

-F'(t)/F(t) <

Jooxe-t~(X)d~Jooe-tXF(X)dX

<

o

0

f

""

-'['x ..

Ir

fco

-'['x -1 ~ € + xe d/{F(€/2) e dxl,.. €+F(e:/2) (€+1/'r)exp[-'['e:/2J, e; €/2

which is less than 2€ for '[' sufficiently large. Hence (2.9) is proved. Final-ly, using these results for the df G(x)

:=

F(x + Y

F), we easily obtain (2.8) for a df F with Y

F > O.

0

The following property of the functions ~A' which we shall use repeatedly, follows directly from (2.8) and (2.9).

Lemma 2.9. Let F be an arbitrary df on [0,00) and 0 ~ A< "", then (cf. Defi-nition 2.6)

(2.10) _{lim ~A('[')}

₌

_{C(YF;A) •}

'['~

In the sequel we use without further comment the following properties of the function KA corresponding to a df F in FA.

Lemma 2.10. Let 0 < A ~ 00 and F E FA (and F(O) > 0 if A = ~), then (cf. De-finition 1.2 and 1.3)

i) If F(x) > 0 for all x > 0, then KA(O) = 0,

co

J

...

ii) dK

A (x) = c" {l - F(A)}.

0

Proof. Part i) of the lemma immediately follows from x F(x)K,,(O)

~

f

F(x - y)dKA(y) ,.. 0-x

f

c(Yj,,)dF(y)

o

if F € FA with

°

< A < co, and, if F E F+, _co from

x

F(x)K~(O)

s

J

F(x -

y)dK~(y) =

F(x) - F(O)

0-Part ii) is obtained from (2.6) and (2.7) by letting t ~ O.

3. The monotonicity of F",absolute continuity

From now on we only consider df's F on [O,~) with the property

(3. 1 ) _{"x>O F(x) > 0 •}

This is not an essential restriction (cf. Theorem 4.2i». It follows that

lim ~,,(T)

=

0 for 0

s "

< ~ (see (2.10», and that K,,(O) = 0, if F E F" and

T-+oo

Os" :s; "".

First we prove that all distributions in the classes FA with 0 < " < "" are inf div.

Theorem 3.1. For all A E (O,~) we have FA C FO'

Proof. Let 0 < " < 00 and F E FA' Then (2.6) holds and can be rewritten as

(3.2)

...

Iterating this equation and using the fact that F(t + nA)/F(n,,) tends to 1

if n ~ 00 (see (2.8», it follows that

(3.3) -1'" 1 - c_A KA(kA) -1... • k=O 1 - c A KA (t +kA) II -I'" A ...

If we define 1Tk :=

c"

KA(kA)

«

1, see Lemma 2.IOH» and Gk(t) := K~(T +kA)/ KA(k,,) (which is a PLST), then (3.3) takes the form

""

F(T)

=

IT (I - 'IT

k)/(1 - 'ITk

G

k(T» k=O

...

and (cf. Theorem 2.4) F is the limit of a sequence of products of compound-geometric PLST's, which are inf div. Hence F is inf div.

0

- 10

-From (3.3) one easily obtains the following representation for df's in FA with a < A < ~ (cf. Theorem 2.1, 2.3 and 2.4).

-Corollary 3.2. Let a < A < ~, then F _{E: FA iffF has the form}

""

....

(3.4) F(.) "" ....

n

- pG(k).)

k=a - peeL + kA)

where a ~ p < I and G is a df with G(a) = a.

Next we turn to the general monotonicity property. We note that if F. has this property, it also holds for F'

.

_{' .}

F+ and

r .

.

The last assertion is jus-tified by the observation that

(3.5)

r

=

{F E: F+

I

F is a lattice df} •

ct -log ct

Theorem 3.3. For all A and ~ 1n [a,~) we have (3.6)

Proof. In view of Lemma 2.7 we have to show that, if a < U $ A < ~ and if

~A is comp mon, then ~~ is comp man. According to the definition of ~~ we have

(3.7) c -1 {~(1:)

~ ~

~

(T + A)}

=

F(T

+ A +

~)

_

FCT

+

~)

~

FC.

+ A)

FCL)

Dividing by P(T + ~), the right-hand side of (3.7) becomes symmetric in A and ~, and so

....

(3.8)

CA{~,,(T)

-

~

(T + A)}

=

F(T + lJ) C

{~A(T)

-

~A(T

+ lJ)} •

~ lJ P(1: + A) lJ

If ~A is camp mon, then, by Bernstein's theorem, ~A(1:) - ~AC1: + lJ) is camp man too. Further, as F E FA c Fa (Theorem 3.1) and lJ $ A, from Theorem 2.5

it follows that the function

Fer

+ ~)/F(L + A)

=

F(L' )lFCT'

+ A - ~) (with T' ;= T + lJ) is comp mont So, from (3.8) we conclude that ~ (1:) - ~ (T + }..)

lJ ~

can write

is a comp mon function. Now, as lim ~ (T

n-+co ~ + nA)

=

0 (Lemma 2.9), for W we ~

{~ (T + kA) - ~ (1: + kA. + A)} •

It follows that ~ is the limit of a sequence of sums of comp mon functions. II

Hence ~ is comp mon, and the theorem is proved.

0

II

By letting A ~ ~ in (3.8) we easily see that if ~oo is comp mon, then ~ll is comp mon for all II € (0,00), so we have F: c

n

F~. On the other hand

X<oo ~oo

=

lim ~A is comp mon, if all ~A are, and

A~

so F+ ..

00

n

F~. This allows us

A<OO to define the classes F and F' as _{00 00} follows (cf. (1.8».

Definition 3.4.

(3.9)

Foo:=

n

FA (= lim

FA'

because of Theorem 3.3) ,

A <00 X~

F' := {F € F F is absolutely continuous} •

00 00

If F €

FA'

then Theorem 3.3 ensures the existence of the nondecreasing

func-tions K (ll ~ X), corresponding to F. From (3.8) we obtain the following il

properties of K • II

Corollary 3.5. If X E (O,ooJ and if the function K

A, corresponding to F E FA' is absolutely continuous (density k

A), then for all II € [O,AJ the function K is also 'absolutely continuous (density k ), and the following inequality

II II

holds

(3. 10)

Proof. First take llE (O,A). As we saw in the proof of Theorem 3.3,

F(r

+ ll)/F(T + A) is comp mon, so there is a nondecreasing function KA '

A A 4 ,ll

zero for negative arguments, such that F(, + lJ)/F(, + A)

=

KX (f). Writing

A A ,il

~A = KA and <PlJ

=

K

lJ, from (3.8) it follows that for all x > 0 x (3.11)

J

_{c(y;A)dK].I(Y)}

₌

°

x x-y

f f

c(u;].I)dKA(u)dKA,j.l(Y) 0-

°

If K).. has a density k

A, then, changing the order of integration in (3.11) and noting that K, (0)

=

lim

F(,

+

ill/Fer

+ A)

=

1, we obtain the absolute

1\,].1

,~

continuity of K , with density k J given by

(3. 12) k (x) l.J

12

-x

-1

r

= C(X;A) {c(x;l.J)kA(x) +

J

C(X-y;l.J)kA(x-y)dKA,l.J(y)} 0+

For the case l.J

=

0, we let l.J ~ 0+ in (3.8) to obtain (3. 13)

which, using (2.6), is equivalent to

(3. 14) x

J

c(y;A)dKO(y)

=

o

x

f

Y dK A (y) +

o

x

J

KA(x-y)(l-e-AY)dKo(y)

o

It follows that KO is absolutely continuous with density kO' given by

(3. 15)

x

f

kA(x-y)O-e-AY)dKo(y)} •

o

Finally, (3.10) is obtained from (3.12) and (3.15). 0

From the definition of FA it follows that the absolute continuity of KA is sufficient for the absolute continuity of Fex) - F(O). This observation ge-neralizes a theorem by Tucker (1962), if restricted to the half-line; in our case the proof is very simple.

Theorem 3.6. If A EO [O,ooJ and i f the function K

A, corresponding to F EO FA' is absolutely continuous, then F(x) - FeO) is absolutely continuous.

Proof. If KA has a density k

A, then the right-hand side of (1.7) is absolu-tely continuous with density u given by

x

u(x)

=

I

kA (x - y)dF(y) •

0-As C(X;A) > 0 for all x > 0, from (1.7) it now follows that x F(x) - F(O)

=

J

dF(y)

=

0+ 0+ x

J

-1 C(y;A) u(y)dy

So, F(x) - F(O) is absolutely continuous with density fO given by x

(3. 16) _{fO(x) '"} C(X;A)

-\ f

kA (x - y)dF(y) •

0-Corollary 3.7. If 0 then F € F~.

+

~ A < ~ and F € FA\FA with absolutely continuous K

A,

Finally, we give a characterization of F; (cf. (1.5», which easily follows from the definition of FA'

o

Theorem 3.8. Let 0 ~ A <

ro,

then a pdf f is in F~ iff there is a nondecreas-ing function KA such that almost everywhere

(3. 17)

x

c(x,A)f(x)

~

J

f(x - y)dK

A (y) •

o

4. Properties of FA' examples

In this section we shall frequently use the characterization of FA given by Lemma 2.7. For notational convenience we denote the functions ~A' correspond-ing to a df

Fv

(see Definition 2.6), by

~;v).

We start with some properties of the classes FA' The first of them is well known for FO (see [5J).

Theorem 4.1. For 0 ~ A ~ ~ the class FA is closed under weak convergence, i.e. a df F, for which there are Fn € FA such that F(T) '" lim Fn(T), is again

n-+<><> in FA'

Proof. In view of (3.9) it is sufficient to consider the case 0 < A < 00.

The functions

~~n),

corresponding to Fn € FA' are camp mon, and as

~in)

+

~A'

if Fn +

P,

it follows that ~A is comp man too. So F E FA'

0

It turns out that every FA (0 ~ A ~ ~) is closed under translationst but

14

-Theorem 4.2. Let

o ::;;

A ::;; QQ and a > 0, then

i) A df F is 1n FA iff the df F (x) := _{F(x - a) is in FA'}

a

ii) _{A df F is in FA iff the df F (x)} := _{F(ax) is in FaA'}

a

Proof. The theorem is known for A = 0 and follows for A

=

QQ, as soon as it

has been proved for finite A's. So let 0 < A < 00 and a > O. In case i) we

.... -aT ....

have F (T)

=

e F(T), from which one easily obtains the following relation a .

(4. 1 )

As

~A

and

~~a)

are

~oth

nonnegative, it follows that

~A

is comp mon iff

~~a)

is comp mono Hence i) is proved.

In case ii), using F (T)

=

F(T/a), one easily verifies that a

(4.2)

It follows that

~A

is comp mon iff

~~~)

is comp mon, and ii) follows.

0

In view of Theorem 4.2ii) for many purposes, such as asymptotic behaviour and properties of moments, it is sufficient to consider only the class Fl in stead of all classes FA for 0 < A < QQ. Still, the monotonicity of the FA

(Theorem 3.3) is an interesting property, and the separate FA are needed to define the class Foo' Also, in the discrete case, where the lattice is kept fixed, the classes

r

are essentially distinct.

a

In the following theorem we state some simple properties of the FA for

o

< A < QQ, which are well known or trivial for A = O.

Theorem 4.3. For 0 < A < (lQ we have

i) _{if F € FA' then F(T +} v)/F(v) e FA (0::;; v < (0), ii) _{if FE FA'} thenF .... v e FA (0 ~ v ::;; I),

n-l

iii) if F E FA' then F (T):= 11 F(-r+kv)/F(kv) eF (O::;;v::;;A/n; n=I;2, •• ),

n,v k=O v

iv) _{if FE FA'} v) a df F is

then Fv(T) := F(v)F(T)/F(T + v) e FA (0 ::;; v < (0),

in FA iff the function F(A)F(T)/F(T + A) is a PLST in

F:.

Proof. In all five cases Lemma 2.7 is applied. We give the proof for the ca-ses ii), iv) and v); the other caca-ses are easily verified.

we obtain the relation

(4.3) A A 1-v d

v[F(,)/F(, + A)J {- dt ~A (t)} •

If F E FA' then the last factor in the right-hand side of (4.3) is comp mono Further, by Theorem 2.5 we know that F(A)F(t)/F(T + A) E F

O' and hence

A A A I-v (v)

[F(A)F(T)/F(T + A)J . E FO for 0 $; v $; 1. Now, as CPA (T) ~ 0, from (4.3)

it follows that

cp~v)

is comp mono

iv). We note that, if F E FA' then Fv is indeed a PLST (Theorem 2.5). So we may calculate the function

cp~v)

for Fv and obtain

(4.4 )

If F E FA' then CPA (,) - CPA (T + v) is comp mon, and using again Theorem 2.5, from (4.4) we conclude that

cp~V)

is comp mono

v). If F € FA' then the function F(A)F(,)/F(T + A) is a PLST FA' say. As

FA (0)

=

F(A) lim F(t)/F(, + A)

=

F(A) (see (2.8», we have t-l>«!

(4.5) ₌₌ 1

It follows that CPA is comp mon iff

cp~A)

is comp mon, so q) is proved.·

0

We note that now part i), ii) and iv) of the preceding theorem also follow for A

=

~, while part iii) takes the form

n-l

(4.6) F E F .. F (T):

=

IT F (t + kv) /F (kv) E F (0 $; v < ~; n

=

1 ,2, •• ) •

~ n,V k=O v

Finally, we prove that U FA is dense in FO in the sense of weak convergence. A>O

Theorem 4.4. I f F € F

O' then there is a. decreasing sequence {An

}7

with An-+O and there are Fn € FA (n

=

1,2, ••• ), such that for every t ~ 0

n

(4.7) FCT) - lim F (,) •

n-l>«! n

+

Proof. On account of Theorem 2.3, for F E FO we can give a proof along the same lines as in [2J for the discrete case. For an arbitrary F E FO this proof and the proof of De Finetti's theorem (see [5J, p. 112) suggest the

-2

- 16 -n-1 .... (4.8) F ('r) == II n k==O _L"n-

i

2 - n 2F (T + kin ) ... which on account of (4.6) is a PLST in

F

-2' We can rewrite Fn as

n

_1 n-]

.... -~ ... n 2 n

F (1) ., {I + n (F (T) - I)} IT [I + €k(n)J ,

n k-O

where the €k satisfy €k(n) ., o(l/n) (n + ~) uniformly in k, as can be proved without too much difficulty. Now it easily follows that

F

(1) + F(T) for

n every T ~ 0 as n + ~.

Next we mention some simple examples of distributions in FA' noting that

in

the following section some more examples for

F

_eo are given.

As

the proofs of the following statements are often rather technical, but not very difficult, they are omitted; we only give very brief indications.

J) By taking F € F+ _eo in (4.6) we get

n-] _ pG(kA)

II € FA (n .. 1 ,2, ••• ,co; 0 S P < 1; G .. df; 0 S A < eo) •

k=O - pG(T + kA)

2) Take G(T) =

~(~

+ T)-l in example 1, then it follows that

n-l ~l + kA ~2 + kA

II k

I

€F (n-I,2, ••• ,eo; 0<]JlS1l2; OSA<CO).

k=O ]J 1 + A + T ]J 2 + kA + T A

3) Calculating the function

<P"" we obtain for positive ]J. l (i '" 1,2,3)

]Jl ]J3 ll2 2

I {

} _{E F"" • !(]Jl} + _{+ ]J3)} :::;; ]J2 S max(]Jt ,]J3) ~1 + T ]J3 + T ]J2 + 1 .... _-1 4) By choosing F(T)

..

1l(]J + T) in (4.6) we get n-I II II + kv _" c

F'

(n

=

1 2 _{J t " ' ;} II _> 0 0 _{; S v < co} ) _• k=O ]J + kv + T V

5) Take n

=

2 in example 4 and calculate <P

A, then

]J ~ + v

F

)

]J + T ]J + V + T € A· A S v (]J > 0; v ~ 0 .

6) Immediately from the preceding example U FA (ll > 0) • A>O

17

-5.

The class

F

OIl

In [7] Steutel proved the infinite divisibility of the comp mon pdf's on (0,·). Le. of mixtures of exponential pdf' •• and. slightly more gener.l, of the log-convex pdf's on (0.·). We now prove that these are in the class.

F!.

Theorem 5.1. If a pdf f on (0,00) ·is log-convex, then. f e F!.

Proof. For h

>

0 define p!h) :_ y

hhf(6(2n + l)h) (n - 0.1.2 •••• ) • where Y

h is such that

I~ p~h)

.. 1. As f is a log-convex function,'

{p!h)}~

is a log-convex sequence, and hence satisfies the recursion relation (1.2), with nonnegative rtl. (0) (see [7]). Considering

{p~h)}~

as a probability

dis-ttibution on {0,h,2h.3h,. •• }. it follows: (ef.

(3.5»

that for all h > 0

(h) OIl . ' + . ' . . .

{Pn 10 E F •• Now,for the df Fj corresponding to. f,.we have

[xlh] (h)

F(x) • lim

2

hi( H2n + 1 )h)- l i m r Pn •

h+O n-O h+O nh~x

As F GO is closed under, weak convergence (Theorem 4.1), we conclude thatF E F

co.D

Corollary 5.2. I f f is a comp mon pc.if on (0,00), then f E F!; or, in terms

of PLST's: for all df's G on (0,00) we have

(5.1)

_{:- f}

o

A

simple example of the situation above is. provided by the gamma-distribu-tion

....

(5.2) F(T)

.... +

We know that if f is a comp mon pdf on (0,00), then p + (I - p)F(T)

eF

0 for

o

< pSI. Now the question arises for which df's F we have p + (1 - p)F(T) E:

~.

It will turn out that this holds for all _{' .} F €

F •

₀₀ To this end we use the f01-lowing characterization of F • _GO which can be considered as an extension of Lemma 2.7ii) from

F:

to

F

••

- 18

-Theorem 5.3. A df F is

~n

F iff

~

F(T)-1 is camp man. "" dT

Proof. In view of Definition 3.4 and Lemma 2.7i) we have to prove that ~A is d .... -1

camp man for all

A

< "" iff ~(T)

:=

r.r F(T) is camp man. If ~ is camp mon, then

d.... - ] . . . . - 1

W(T) - WeT + A)

= -

r.r{F(T + A) - F(T) } is camp man. As F(T + A) ~ F(T), it follows that

is a camp man function, and hence ~A is camp man.

Conversely, if all

~A

are camp man, then so are the functions -

~T ~}"

Cr), which can be written as

Dividing by F(A), and using the fact that if A ~ "", then c}" ~ 1,

F(T + },,)/F(A) ~ 1 (see (2.8» and ~O(T + A) ~

°

(see (2.10», it follows that

(5.3)

and hence is a camp man function.

We note that in the preceding theorem it is essential that F has property

o

... -aT

(3.1); for example, the PLST G(T) := e ~/(~ + T), with a >

a

and ~ > 0, is d ... -1

in F"" «5.2) and Theorem 4.2i», while r.r G(T) is not camp man. As Theorem 5.3 is used for the following theorems, (3.1) is also essential there.

Theorem 5.4. If F E F"" and p ;::: 0, then F(T)/{p + (1 -p)F(T)} is a PLST in F"".

Proof. Defining q>(T) := F(T)/{p + (l-p)F(-c)}, we can write

-I ... -1

is a comp mon function, and as also $(1) ~ 0, from a theorem of Feller (see [IJ, p. 441) it follows that ~ is comp mont Now, as ~(O) = 1, there is a df

A d A -1

G such that ~

=

G, while --d _~ G(~) is comp mont So (Theorem 5.3) G E F [

00

Theorem 5.5. If F ~ F~ and 0 < p ~ 1, then G(~) ;= p + (l -p)F(~). ~ ~. .... F+

Proof. We calculate the function ~~ corresponding to G, and obtain

cpoo(~).

1 - p+(l-p)F(O);:: (l-p){l-:(O)}

~('r)

....

p + (I -p)F(~) F('r) p + (1 -p)F(or)

Using the preceding theorem (and Lemma 2.7ii), if F(O) > 0), it now follows that cp~ is comp mon, and, as G(O) ~

P

>

0,

by Lemma 2.7ii) we may conclude

that G € F+ _~. 0

Corollary 5.6.

i) If a df F on [o,~) has a log-convex density and if 0 < P ~ 1, then (5.4 ) p + (1 - p)i(or) ~ F+ • _~

ii) For all p € (0,1] and all df's G on (O,~)

(5.5) p + (1 - p)

f

~(~

+

T)-ldG(~) ~

F: .

o

As an example we have

(5.6) .... ( ) _{F T :- v + T} v / _~ ~ _{+ T} e F+ ( _{~ 0 < v ~ ~} ) _, as

F

can be put in the form (5.5).

From the correspondence between

r

0 and the r.enewal sequences (see the remark in section 1) and Theorem 5.5, we easily obtain the following property of re-newal sequences.

00 00

Corollary 5.7. If {un}O ~s a renewal sequence, then so is {vn}O' where

20

-Next we mention another subset of F • This subset was suggested by a special

""

class of p-functions (see [4]). Distributions of this type also occur as pas-sage-time distributions in [6].

Theorem 5.8. If p E [0,1), ~ >

°

and v > 0, with v S ~, and if G is a df on

(0,""), then

(5.8) F (1') : .... = -~-II - - - -.... 1 - P - - _ _ . - - E F ~ •

~ + l' 1 - p{v/(v + T)}G(1')

Proof. In view of Theorem 5.3 we calculate

d .... -1 d v A

\l (I - p)d'T F ( T ) =

F

(ll + T) (1 - p v + -r G ( T ) )

J

=

which, looking at (5.6), for v S ~ is camp mono Hence F E

F"",

while F is

ab-solutely continuous because of the exponential component.

o

By taking G(T) - in (5.8) we get the following example

(5.9)

The p-functions are closely related to the pdf's in F",,; in fact, a

represen-A

tation theorem can be derived for F E

F"",

very similar to that for LT's of p-functions (cf. [4J, p. 55)1. We shall report on this in detail later, and restrict ourselves here to a representation theorem for

Foo,

which can easi-ly be obtained from Theorem 5.3. From this representation theorem it follows without much difficulty that (save for norming) Foo contains both the renewal sequences and the standard p-functions.

Theorem 5.9. If F is a df in F , then _oo

F

has the form (5. 10)

where h is such that

(5. I I ) exp[-h(-r)J is a PLST in

FO •

Conversely, every function h, satisfying (5.1]), defines by (5.10) a PLST

F

in

F •

""

A similar representation theorem has recently been proved by J. Hawkes for potential kernels in Levy-processes (oral communication).

Proof. Let F be in F , and define the function h by ₀₀ .... -I

her) := F(,) - I •

Then (5.10) holds, and as h(O)

=

0, he,)

~

0 and

~,

he,) is comp mon (Theo-rem 5.3), it follows (see [IJ, p. 441) that exp[-h(,)J is a PLST

G,

say. Clearly, the function ~O' corresponding to G, is comp mon, and so G E F

O' ....

Conversely, let h be a function such that exp[-h(,)J is a PLST G in FO' Then

.... d d ....

he,)

=

-log G(,) ~ 0 and

dT

he,)

= -

dT

log G(,) is comp mon, so (see [IJ,

-1 .... d.... -1 d

p. 441) {I + he,)} is a PLST F. As

dT

F(,)

=

d, he,) is comp mon, we

have F E F •

0

00

We conclude this section with a simple application of Theorem 5.9; for all a > 0 and v > 0 we have

(5. 12) F(,) := {I + a 10g(1 + VT)}-I E F

00

Acknowledgement. I would like to thank F.W. Steutel for suggesting the pro-blem and for many helpful discussions.

22

-References

.[IJ Feller, W. (1971). An introduction to probability theory and its applications 2, 2nd. ed. Wiley, New York.

[2J Harn, K. van and Steutel, F.W. (1976). Generalized renewal sequences and infinitely divisible lattice distributions. Stochastic Pro-cesses and their Applications 5 01-09.

[3J Katti, S.K. (1967). Infinite divisibility of integer valued random

variables. Ann. Math. Statist. 38 1306-1308.

[4J Kingman, J.F.C. (1972). Regenerative phenomena. Wiley, London.

[5J Lukacs, E. (1970). Characteristic functions, 2nd. ed. Griffin, London.

[6J Miller, H.D. (1967). A note on passage times and infinitely divisible distributions. J. Appl. Probe 4 402-405.

[7J Steutel, F.W. (1970). Preservation of infinite divisihility under mixing, and related topics. Math. Centre Tracts 33. Math. Centre, Amsterdam.

[8J Steutel, F.W. (1971). On the zeros of infinitely divisible densities. Ann. Math. Statist. 42 812-815.

[9J Tucker, H.G. (1962). Absolute continuity of infinitely divisible