Discrete analogues of self-decomposability and stability

Discrete analogues of self-decomposability and stability

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Steutel, F. W., & Harn, van, K. (1978). Discrete analogues of self-decomposability and stability. (Memorandum COSOR; Vol. 7807). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 78-07

Discrete analogues of self-decomposability and stability

by

F.W. Steutel and K. van Barn

Eindhoven, March 1978

Abstract

Analogues are proposed for the concepts of self-decomposability and stability for distributions on the nonnegative integers. It turns out

that these "discrete self-decomposable" and "discrete stable" distributions have properties that are quite similar to those of their continuous counter-parts.

AMS(MOS) Subject classifications (1970): Primary 60E05, 60F05

Key words and phrases: discrete distribution, self-decomposable, stable, infinitely divisible, unimodal, domain of attraction.

Discrete analogues of self-decomposability and stability

By F.W. Steutel and K. van Harn

TeohnoLogioaZ Univer8ity, Eindhoven, The NetherLand8 1. Introduction and preliminaries

A probability distribution on R is said to be self-decomposable (or, of class L) if its characteristic function (c.f.) satisfies (cf. [5J, p. 161)

(1.1) ~(t)

=

~(at) ~ (t)

a (t E R; a E (0, 1» ,

with ~ a c.f. For the corresponding random variables (r.v.'s) this means a

that "in distribution)

( 1.2)

x

= aX' + X

a (aE (0,1»,

where X' and X are independent and X'is distributed as X. Clearly, apart a

from X= 0, no lattice r.v. can satisfy (1.2); in fact, all nondegenerate self-decomposable (self-dec) distributions are known to be absolutely continuous (see e.g. [3J).

In this note we propose analogues of self-decomposability and stability· for distributions on :IN

° :

=

{o,

1,2, ••• }. It turns out that the discrete self-dec distributions and the discrete stable distributions share the basic properties with their continuous counterparts. The discrete self-dec

distributions, for i~stance, are unimodal, and the discrete stable distributions are very similar to their continuous analogues on (O,w).

We shall need the following two lemmas for probability generating functions (p.g.f.'s), and on infinite divisibility (inf div). For a proof of the

second lemma we refer to [IJ and [6J. The generating function of sequences

w 00

(an)O' (bn)O' etc. will be denoted by A, B, etc.

Lemma 1.1. If P is a p.g.f., then

lim (1 - x) P'(x)

=

0.

xtl

Proof. For x E [0,1) we have 1 - P(x) = (1 - x) P'(~) with ~ E (x, 1). As p' is nondecreasing, we have (1 - x) P'(x) $ (1 - x) P'(~)

=

1 - P(x) +

°

as

2

-Lemma 1.2. A p.g.f. P with 0 < Po < I is inf div iff P has the form

(1.3) P(z)

=

exp{A(G(Z) - I)},

where A > 0 and G is a (unique) p.g.f with G(O) = O. Equivalently, P is inf div iff

( 1.4) _{F{z) - exp{ -}

J

R(u)du},

z

where R(u)

₌

~ E r un, with r ~ 0 and, necessarily, ~ E r (n+l) -1

o n n o n

iff the p satisfy

n ( 1.5) (n + l)p 1

=

n+ with r ~ O. n n

L

Pkr-k k=O n 2. Discret~ self-decomposability i.e.

We start with analogues of (1.1) and (1.2) that operate within the set of distributions on ::N O. For definiteness we shall assume 0 < Po < 1.

Definition 2.1. A dis tribution on ::N 0 with p.g. f. P is called discrete self-decomposable if (2. I) P(z)

=

pel - a + az)P (z) a with P a p.g.f. a

(Izi

~ 1; \:L € (0,1),

Eq~ation (2.1) can be written ~n terms of r.v.'s as follows:

(2.2)

x

=

a 0 Xl + X ,

a

where a 0 X' and X are independent, and Xl is distributed as X. Here

a

a 0 X is defined (in distribution) by its p.g.f. P(l - a + az), or by

X (2.3) a 0 X =

I

N. ,

1 J

where peN.

=

1)

=

I - peN.

=

0) = a, all r.v.'s being independent. It then

J J

follows that a 0 X E ::NO' with loX = X, 0 0 X

=

0, and E a 0 X

=

a E X,

as in scalar multiplication; an empty sum is zero.

We first establish the canonical form of the discrete self-decomposable p.g.f.'s.

3

-Theorem 2.2. A p.g.f. P is discrete self-dec iff it has the form

1

(2.4) P(z)

I

I-G(u)

= exp{-A 1 _ u du},

z

where A > 0 and G is a (unique) p.g.f. with G(O)

=

O. Equivalently, P is discrete self-dec iff it is inf div and has a canonical measure r _n

(cf. lemma 1.2) that is nonincreasing.

Proof. Let P be s.d., Le. satisfy (2.1). Then for r > 0 and r(l - d )-1 E :N,

n (2.5) Q (z):= r,n {p 0. n r/( 1-0. ) (z)} n is a p.g.f. As pel - 0. +

o.z)

=

P(z) + (1 - 0.)(1 - z)P'(z) + 0(1 -

a)

as 0. t 1, by (2.1) and (2.5), with 0. such that 0. t 1 as n + ~,

n n

(2.6) Q (z) := lim Q (z)

=

exp{-r(1 - z)P'(z)/P(z)}.

r n+m r,n

As (cf. lemma 1.1) Q (z) + 1 as z t 1, by the continuity theorem for p.g.f.'s

r

(cf. [IJ, p.280), Q

r is a p.g.f. for every r> O. It follows that Q := Q1 is

infinitely divisible, and therefore by (2.6), and (1.3) applied to Q, that

(2.7) R(z) _:= pI (z) _P(z)

_{= -}

log Q(z)

=

i-z

A1-G(z) }-z '

equivalent to (2.4). Comparing (2.4) and (1.4) we see that P is inf div, with (2.8) r = A(i -n n

I

j=i g.) == A J ~

L

j=n+l g. , J

which is nonincreasing. Conversely, let P satisfy (2.4); this is easily seen to be the case if P is inf div with nonincreasing r , i.e. satisfying

n

(2.8). Then P satisfies (2.1) with

l-o.JI-Z)

Po.(z)

=

exp{- R(u)du},

4

-L.e. with R (z)

:=

P'(z)/P (z)

=

R(z) - a R(l-a(l-z)), with coefficients

a a r - a n {I -n+l a 00

L

(njj)(l - a)j}

=

0, j=O

where we have used the fact that r is nonincreasing. It follows that P

n a

is a (infinitely divisible) p.g.f.

The unimodality of discrete self-decomposable distributions is a corollary to the following theorem.

00 00

Theorem 2.3. Let (Pn)O and (rn)O be sequences of real numbers with Pn ~ 0, PO > 0, and rn nonincreasing. Furthermore let Pn and rn be related by

(2.9) _{(n + 1) P n+ 1 =}

n

L

Pk rn-k k=O

Then

(pn)~

is unimodaZ, i.e. P

n - Pn-I changes sign at most once (p-1 = 0); P n is nonincreasing iff rO ~ 1.

Proof. The proof is very similar to that in [7J for self-decomposable densities on (0,00). Putting d = P - P and A = r - r l' from (2.9)

n n n-l n n n+

we obtain by subtraction

(2.10)

Clearly, d

n ~ 0 for n E ]N iff rO ~ 1. Now let rO > 1, and suppose that (2.11) . d 1 > 0, d

2 ~ 0, ••• ,

Then we have, putting Pn-j = 0 if J > n,

P n -j ~ _{P (j = m+l, m+2, ••• )} (2. 12) 1 n 2-J P n -j ~ P (j = 1,2, ••• ,m)

.

1 n1

From (2.10) and (2.11) we have

n -1 1

(2.13) (n

1 + l)d n 1 = (r - I)p -

L

A . Pn -j-l < 0 ,

5 -n -1 Z (Z.14) (n₂ + l)d + n Z = (r O - l)p n

-Z

j=O

I

A. J p n Z -J' -1 > O. m-l m-} n-} A. )

As 2: A.p ::; _{E A.p . 1}

,

from (2.14) it follows that (r

O

=

r + E

0 _{J n2} 0 J n -J-₂ n j=O J

n -1

(2.15) (r - l)p _{m n} > ,E 1 A. _{Pn -j-l}

.

2 J=m J 2

But, from (2.12) and (2.15) we obtain n -1

1

m-I

n -1 1

l:

j=O A, J Pn 1 -J'-1 ::; j=O

l:

A. J p n_l + j=m

l

which contradicts (2.13). It follows that (2.11) is impossible.

Corollary 2.4. A discrete creasing iff rO

=

PI/PO ::;

(with Po > 0) is unimodal

00

self-dec distribution (Pn)O is unimodal; it is

nonin-I. Equivalently, an inf div distribution on mO

if r (cf. (1.5» is nonincreasing; it is non-n

increasing iff in addition rO ::; 1.

Remark 1. In theorem 2.3 the r are not supposed to be all nonnegative, i.e. we n

seem to find a sufficient condition for unimodality of more general sequences than inf div distributions. For nonnegative p , however, r nonincreasing

n n

imp lies r n ;::: 0 (n €

m

0) •

Remark 2. Theorem 2.3 could be used to give a slightly simpler proof of the unimodality of continuous self-dec distributions on (0,00), as any such

distribution is the limit of discrete self-dec distributions. This procedure amounts to a more drastic discretization than the one used in [7J.

3. Discrete stability

The set of distributions on B that are (strictly) stable with exponent y is the subset of the set of self-decomposable distributions with r.v.'s X satisfying (cf.[2J, p. 171)

(3.1) (s,t> 0),

in distribution, where XI and X

2 are independent and distributed as X. We rewri te (3. 1) as

6

-Now replacing aX

I by a 0 XI as defined in (2.3), and similarly for the other term, we obtain the discrete analogue of (3.2). In terms of p.g.f.'s we then have

(3.3) P(z) = pel - a(l-z» P(l - (1 - aY)l/Y(l - z» (Izl s 1 a E (0,1»,

and we give the following definition.

Definition 3.1. A p.g.f. P (with 0 < P(O) < I) is called (strictly) discrete stable with exponent Y > 0 if it satisfies (3.3).

From (3.3) it follows that

pel - a(l-z» - P(z) + P'(z)

=

(1 - a)(l - z)P(1 - a(l-z» P(z) as a t 1. Putting (1 - aY)I/y

=

_{u, this means that}

(3.4) - P (1 - u ( l-z» . + y- I ( 1 ) I -y p' (z) , - z P(z) (u(l-z» y and with Z =0, (3.5) 1 - P(I - u) uY

Combining (3.4) and (3.5) we conclude that

(3.6) p' (z) P(z)

(u ... 0),

(u '" 0).

(z E [0,1).

As pl(l) > 0 (possibly infinite) unless P(O)

=

I, from (3.6) we see that

o

< y's 1.

Integrating (3.6) we obtain

(3.7) P(z)

=

P~(z)

:= exp{-A(I - z)y}

(izi s

1 ; A > 0),

by analytic continuation. As any P satisfying (3.7) satisfies (3.3), we have now proved

Theorem 3.2. Discrete stable p.g.f.'s (i.e. satisfying (3.3» only exist for y E (O,IJ, and all stable p.g.f.ls with exponent yare given by (3.7).

Remark. The discrete stable p.g.f.'s are quite similar to the Laplace

transforms exp(-ATY) of the stable distributions on (0,00) (cf.[2J, p. 448). Rather curiously, the Poisson distribution replaces the degenerate one,

7

-i.e. we have

Corollary 3.3. The Poisson distribution is discrete stable with exponent one.

Further, as in the continuous case, we have by (3.3) and (2.1)

Corollary 3.4. A discrete stable distribution is discrete self-decomposable, and hence unimodal.

Remark. If we define a p.g.f. P to be in the domain of (discrete) attraction of a stable p.g.f. P

y if there exist an such that

lim {p (1 - a + a z)} n = P ( z)

n-+<>o n n y

then it follows that all distributions with finite first moment are attracted

-1

by the Poisson distribution: take a

=

n • A general theory of attraction

n

could easily be developed. However, as for y E (0,1) we have P_y(1 - ~)

=

exp(-~Y), and for every finite ~ ~

°

pn(l - a ~)

=

{E exp(X 10g(1 - a ~»}n ~ {E exp(-a .X)}n (n + 00),

n n n

X E IN

°

is in the domain of discrete attraction of

P~

iff it is in the domain of attraction of exp(-A. Y) (cf. remark following theorem 3.2).

4. Concluding remarks

We were led to consider equation (2.1) by first considering a~re fQrmal analogue of (1.1), viz. (cf.[4])

(4.1)

(Izl

:s; 1,

a

E ,0,1».

This equation can be treated in the same way as (2.1), and it turns out that one has

Theorem 4.1. A p.g.f. P, with P(O) > 0, satisfies (4.1) iff it is infinitely divisible, i.e. (cf. (1.3» iff it is compound Poisson.

Defining a

*

X (in distribution) by its p.g.f. 1 - a + aP(z), or by

N

a

*

X

=

I

1

X. ,

8

-with N as in (2.3), we may consider the equation X

=

1n terms of p.g.f.'s a.

*

X' + X , or a. (4.2) P(z) = {I - a. + a. P(z)} P (z) ct

(Izl

~ a. E (0,1), to obtain

Theorem 4.2. A p.g.f. P, with P(O) > 0, satisfies (4.2) iff it is compound geometric.

Equation (1.1) can be handled in a similar fashion, evoiding the use of triangular arrays, and one finds in exactly the same way: ~ satisfies (1.1) iff (this seems to be new)

~(t)

• exp

I

h(u)u-1du

°

(t E :R),

where exp(h(u» is an inf div characteristic function. To prove this, however, one needs to know that ~' exists in :R\ {O}, and is such that

t~'(t) +

°

as t + 0.

No such complication arises in the case of distributions on [0,00) if one uses Laplace transforms instead of cof.'s.

Corollary 3.3 seems to suggest that the distribution of a sum of i.i.d. random variables with only a first moment should be approximated by a discrete stable Poisson distribution rather than by a stable degenerate distribution. If higher moments exist, a normal approximation would, of course, be preferable.

It might be possible to develop a theory of disorete limiting distributions for maxima of i.i.d. random variables in EO' This will be investigated later.

[IJ Feller, W. (1968). An introduction to probability theory and its applica-tions, vol. 1, 3-d ed. Wiley, New York.

[2J Feller, W. (1971). An introduction to probability theory and its applica-tions, vol. 2, 2-nd ed. Wiley, New York.

[3J Fisz, M. and Varadarajan, V.S. (1963). A condition for the absolute

continuity of infinitely divisible distributions. Z. Wahrscheinlichkeits-theorie and verw. Gebiete

l'

335-339.

[4J van Ham, K. and Steutel, F.W. (1977). Generalized renewal sequences and infinitely divisible lattice distributions. Stochastic Processes and their Applications ~, 47-55.

9

-[5J Lukacs, E. (1970). Characteristic functions, 2-nd ed. Griffin, London. [6J Steutel, F.W. (1971). On the zeros of infinitely divisible densities.

Ann. Math. Statist. ~, 812-815.

[7J Wolfe, S.J. (1971). On the unimodality of L functions. Ann. Math. Statist. 42, 912-918.