derivations of the compound poisson distribution and process

Statistics & Probability Letters 18 (1993) l-7 North-Holland

30 August 1993

Derivations of the compound Poisson distribution and process

Y.H. Wang Concordia University, Montreal, Que., Canada

Shuixin Ji Northwest Normal University, Lanzhoy China

Received April 1992 Revised November 1992

Abstract: For a sequence of independent integer-valued random variables (X,,}, we give a direct proof of the convergence of the distribution of S, = X,, + . * + X,,, to a compound Poisson distribution with discrete compounding distribution. In our proof, we trace the change in the asymptotical behavior of S, from the Poisson to the compound Poisson distribution as the range of {X,3 is expanded from {0, l} to general subsets of the integers. The corresponding results for the axiomatic derivations of the compound Poisson processes are also obtained.

Keyworuk: Sum of random variables; limit distribution; Poisson; compound Poisson; stochastic process; characterization.

1. Introduction

Let {X$ i = 1,. . .) n} be it independent random variables taking values in a set D C (0, * 1, f 2,. . .I withOED and Sn=Xnl+ *.’ +Xnn. If {X,J are Bernoulli (D = (0, l)), then it is well known that the distribution of S, converges to the Poisson distribution with parameter A if and only if

n xP(Xni=l)+A and l~lynP(X,i=l)-+O (asn+a).

i=l . . (1)

(The ‘if part of the above result was obtained by von Mises (1921) and the other part by Koopman (19.50). We shall call the first part von Mises theorem and the whole thing von Mises-Koopman theorem in the sequel.)

Recently there have been some attempts to generalize the above result to the case that, under the umbrella of independence, IX,,) take values in a set D larger than (0, 11. (See PCrez-Abreu (1991, 1992) and Wang (1989, 1992j.J The most general result in this direction is possibly Theorem 3 in Wang (1989)

Correspondence to: Prof. Y.H. Wang, Department of Mathematics Concordia University, 1455 W. de Maisonneuve Blvd., Montreal, Que., Canada H3G IMS.

0167-7152/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved 1

Volume 18, Number 1 STATISTICS & PROBABILITY LETTERS 30 August 1993

which says that if

~P(Xn,=l)-,A>O (n-m), i=l

2P2(Xni=1) -+o (n-m), P) i=l

&qX,,+oor 1) -to (n+m), (2c) i=l

then the limiting distribution of S, is Poisson with parameter A. It is easy to verify that the conditions (2a) and (2b) together are equivalent to the two conditions in (1)

if {XJ are Bernoulli. Therefore in the extension of von Mises theorem to the general integer-valued case the required additional condition is (2~). (Perez-Abreu (1991) also used the same three conditions.) It is then logical to ask “What would be the limiting distribution of S, if the sum in (2~) converges to a non-zero number?”

The first answer can be found in Geiringer (1960). She considered the case of D = (0, 1,. . . , m} where m z 2 is a positive finite integer and proved that the limiting distribution S, is a compound Poisson. (Geiringer did not look at the problem from our point of view, but was mainly interested in extending von Mises theorem to a sequence of random variables {XJ with range D = (0, 1,. . . , m}, m a 2.) A more complete result in this direction is Theorem 7 in Wang (1989). In that it was proved that if (2~) converges to a nonnegative number then the limiting distribution of S, is a compound Poisson.

The characteristic function technique was used in both Geiringer’s and Wang’s proofs. While the proofs are impeccable, they do not provide any clues as to how the change in the sequence (X,J from Bernoulli to more general integer-valued random variables affects the change in the limiting distribution of S, from Poisson to compound Poisson. In Section 2, we shall give a new proof which relates these, changes.

Parallel to the development of the Poisson convergence theorem is the development of the axiomatic derivation of the Poisson process. Let {X(t): t 2 01 be a counting process with homogeneous independent increments. The most common method of deriving the homogeneous Poisson process today is to solve the family of the differential equations:

i

P;(t) = -A&(l) +2&1(t) (k > l),

p&(t) = -A&(t), (3)

where P&) = p(X(t> = k). This method was proposed more than ninety years ago by Bateman (1910) using the following axioms:

X(0) = 0, (4a)

{X(t)} is homogeneous and has independent increments, (4b)

P(X(h) > 2) =%(h), (4c)

P(X(h) = 1) =Ah + o(h). (4d)

Assuming that {XJ are identically distributed, we can rewrite (1) as

P(Xni= 1) =A(l/n) +0(1/n) for all i= 1, 2,...,n,

and noting that axioms (4a) and (4~) together imply that the increments X(t + h) -X(t) are essentially Bernoulli random variables, we see that Bateman’s four axioms for a counting process (X(t)1 are the rephrasing of the two conditions in (1) for a sequence (XJ of independent and identically distributed

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Bernoulli random variables. In Section 3, we extend Bateman’s theorem to obtain axiomatic derivations of the compound Poisson process with discrete as well as continuous compounding distribution.

2. A compound Poisson convergence theorem

The distribution of Y is say tosbe compound Poisson with parameter A and compounding distribution F if

y= dl’ .*. +zhi i

if N> 1,

if N=O,

where {Z,} are independent and identically distributed random variables with common distribution F and independent of N which is Poisson with parameter A. If F is degenerate at k # 0, then Y = kN. This fact motivates the following lemma which is a characteriiation of the discrete compound Poisson distribution. This characterization theorem nicely relates the discrete compound Poisson distribution with the Poisson distribution. We shall present it here without proof. It can be easily proved by using the generating functions.

Lemma 1. An integer-valued random variable Y has a compound Poissan distribution with parameter h and compounding distribution {A,/h: k ~g}, where B E { f 1, f 2,. . .} and A = Ck l sAk, if and only if, there exists a sequence of independent Poisson random variables (Y,: k C&S} with parameters {A,: k E_@) such that Y= CkE9 kY, (in distribution). q

The sufficient part of the next theorem is Theorem 7 in Wang (19891, but here we provide a new proof. Denote D, = D \ (0) and

en, = 1 -P,,(O), A,,= kP(X,,=k), ‘r~ = c Ank. i=l keD,

Theorem 2. Let {Xni: i = 1,. . . , n) be n independent random variables taking values in a set D G IO, k 1, + 2,. . . } with 0 ED. For all non-empty subsets A of D,, the limiting distribution of

S,,= c k&(X,,i=k) kEA i=l

(5)

is a compound Poisson distribution with parameter A, = C k E AAk and compounding distribution (A,/A,: k E A} if and only if

A,+A>O, &+O, A,, + A, > 0 (for all k ED,) (6) i=l

as n --) 03.

Proof. First we prove the ‘if part: Fix A ED, and let B be an arbitrary non-empty subset of A. Then {1(X,, E B)} is a sequence of independent Bernoulli random variables with

P(I(X&B)=l)=P(X&B)= c P(Xni=k). kcB

Volume 18, Number 1 STATISTICS & PROBABILITY LE’I-I’ERS 30 August 1993

By assumptions,

tP(Z(XniEB)=l)= C Ank+ CA, (asn+m), i=l kcB k=B

and

ic.P2(Z(X,,EB)=1)= e ( C P(Xni=k)}2< ee$+O (as ndrn). i=l kcB i=l

Consequently, by von Mises theorem,

~(z(x~iEB)=k))=e~(-k~BAk)(k~~Ak)k,k!. i=l

By the additivity of the Poisson distribution, the right-hand side of (7) is the distribution of Ck EBYk, where (Yk: k E B} are independent Poisson random variables with parameters {A,:- k E B}.

Since B is an arbitrary subset of A, the ‘if part follows from Lemma 1 and equation (5). Next we prove the ‘only if part: In view of (5) and Lemma 1, the convergence of the distribution of

S,, to a compound Poisson with parameter A, = Ck E A A k and compounding distribution {A,/A,: k EA} for all A CD, implies the convergence of the distributions of

S,, = 5 Z( X,,i = k) (for all k ED,) and T,= kZ(X/O) i-l i=l

to Poisson with parameter A, and A, respectively. Thus by the von Mises-Koopman theorem we have A nk d ‘k and mml,ian P(X,, = k) -+ 0 (n + m>; and A,, + A and maxlGiG,{P(X,,i # 0)) + 0 (n + CO>. The proof is complete by noting

Since SnDo = S,, the next Corollary follows immediately from Theorem 2.

Corollary 3. Zf a sequence of random variables (XJ taking values in a set D c (0, f 1, f 2,. . . } with 0 E D satisfies the three conditions in (6), then the limiting distribution of S, =X,,, + * * - +X,,, is a compound Poisson distribution with parameter A and compounding distribution {A,/A: k E Do}. 0

3. Axiomatic derivation of the compound Poisson process

3.1. The discrete compound Pot&son process

We first give a definition of the compound Poisson process.

Definition 4. A stochastic process (X(t): t & 0) with state space S is said to be a compound Poisson process with parameter m(t) = J,‘A(s) ds for some non-decreasing non-negative function A(t) and a compounding distribution F if

X(t) = i

Z, + .-- +ZNoj if N(tj a 1, o

if N(t) = 0,

Volume 18, Number 1 STATISTICS & PROBABILITY LETI’ERS 30 August 1993

where {Z,} are independent and identically distributed random variables with common distribution F and independent of the Poisson random variable N(t) with m(t) = E(X(t)). The process is said to be homogeneous if and only if m(t) = At, A > 0, for all t z 0. Otherwise it is said to be non-homogeneous.

Since the Poisson process {N(t)) has independent increments and Zi are independent, the compound Poisson process has also independent increments.

It is well-known that the four axioms in (4) for the non-homogeneous Poisson process are (see Ross, 1985):

X(0) = 0, (8a)

{X(t)} has independent increments, (8b)

P(X(t+h) -X(t) z2) =0(h), (8c)

P(X(t+h)-X(t)=l)=h(t)h+o(h). (8d)

The Poisson process derived from (8) has parameter m(t) = /$(s) ds. In the four axioms in (8) the one with room for improvement is evidently the axiom (8~). The next

theorem is a generalization of the Poisson process in this direction. It can be proved directly by using the four axioms in (9) to derive a family of differential equations for p&) = P(X(t) = k) and then solve it, as in the derivation of the continuous compound Poisson process in Theorem 6 in the next section, but we shall follow the proof of Theorem 2. It is more direct and offers a different view of the compound Poisson process.

Theorem 5. If {X(t): t 2 0) is a discrete stochastic process with state space S G (0, f 1, f 2,. . .) sutisfuing

X(0) = 0, (9a)

{X(t)} has independent increments, (9b)

P(X(t+h) -X(t) =k) =o,A(t)h+o(h),

where a,>0 forallk~~~CS\(O) with c (Y -1, kc%3 k-

A(t) z 0 for all t 2 0 and m(t) = k’A(s) ds, (9c)

P(X(t+h) -X(t) =k) =o(h) forkG9und ~0, (94

then X(t) is a compound Poisson process with parameter m(t) and compounding distribution {a,: k ES}.

Proof. Let {X,(t); k E@ be a family of independent counting processes each satisfying the four axioms in (8) with obvious modification of (8d) as

P(X,(t+h)-Xk(t)=l)=okA(t)h+o(h).

Then, by Bateman’s theorem, for each fixed k ~9, {Xk(t)) is a Poisson process with parameter akm(t>. It is easy to verify that if a discrete stochastic process {X(t)) possesses the four axioms in (91, then

x0> = c k EakXk(t> (in distribution). By Lemma 1, the stochastic process {X(t)) is a discrete compound Poisson process with parameter m(t) and compounding distribution (ak: k E9). 0

Here is a simple example: Let 0 <cr < 1, A > 0, A_,(t) = 2Atcu and A,(t) = 2At(l -a) with the remaining A,(t) = 0 for all t 2 0. Then by Theorem 5 {X(t)) is a non-homogeneous compound Poisson process with parameter m(t) = At 2 and compounding distribution P(Z = - 1) = (Y and P(Z = 3) = 1 - CL

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3.2. The continuous compound Poisson processes

The stochastic process {X(t)} we consider in this section has continuous compounding distribution @ with @’ = C#I and state space S = R, the set of real numbers. Denote f(x; t, w) to be the probability density function of X(t) -X(w) for t > w 2 0.

Theorem 6. If {X(t): t 2 0) is a continuous stochastic process satisfying

X(0) = 0,

{ X( t ) } has independent increments,

f(x; t+h, t) =$(x)h(t)h+o(h),

( 1Oa)

( lob)

where C#J( x) , x # 0, is a continuous probability density function and

h(t) >O forall t>O, (1Oc)

f(0; t+h, t)=l-A(t)h+o(h), (104

then X(t) is a non-homogeneous compound Poisson process with parameter m(t) = j,‘A(s) ds and compounding distribution 4(x).

Proof. For brevity we shall write f<x; t) for f(x; t, 0). Denote

g( s; t) = lRf (x; t) eisx dx

to be the characteristic function of f(x; t>. The stochastic process {X(t)} defined by (10) has one atom at X(t) = 0 for all t 2 0. Therefore its

characteristic function has the expression

g(s; t) =f(O; t) +L,(O,eixrf(x; t) dx. (11)

Fix t > 0. If x = 0 in the interval (0, t + h], then x = 0 in both intervals (0, tl and 0, t + hl with probability one. So that

f(0; t+h)=f(O; t)[l-h(t)h] -to(h). ( 12a)

If x # 0 in the interval (0, t + h], then either x in (0, t I and 0 in 0, t + hl, or x - y in (0, t 1 and Y in

(t, t + h] for y f 0, or 0 in (0, t] and x in (t, t + hl with probability one. Therefore

f(x; t +h) =f( x; t)[l -h(t)h] +A(t)hjR\(Olf(x -y; t>+(Y) dy

+f(O; t)+(x)A(t)h + o(h).

From (12a) and (12b), it follows that for all x E R and t > 0,

(12b)

;f(x; t) = --h(t)f(x; t) +A(t)/&(x-Y; t)W) dy+A(t)f(O; t>4(x). (13)

(If x = 0, the last two terms on the right-hand side of (13) are taken to be 0.)

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Multiplying both sides of (13) by eisx and integrating over R on x we get by (11) that

$(s, f) = -A(f)g(s, t> +A(t)g(s, t)@(s) =g(s, t)W)[@(s) - 117 (14)

where @J(S) = jR,cO+$(x> eisx dx is the characteristic function of c$(x>. Theorem 6 thus follows from (14). q

Acknowledgement

The authors would like to thank Professor Zheng-yan Lin for pointing out an ambiguity in the previous proof of Theorem 2.

References

Bateman, H. (19101, Note on the probability distribution of a-particles, Philos. Mag. 20, 704-707.

Geiringer, H. (1960), On a limit theorem leading to a compound Poisson distribution, Math. Z. 72, 229-234.

Koopman, B.O. (1950), Necessary and sufficient, conditions for Poisson’s distribution, Proc. Amer. Math. Sot. 1, 813- 823.

Perez-Abreu, V. (19911, Poisson approximation to power se- ries distributions, Amer. Star&. 45, 42-45.

Perez-Abreu, V. (1992), Letter to the editor, Amer. Statist. 46, 77.

Ross, S.M. (1985), Introduction to Probability Models (Academic Press, New York, 3rd ed.).

Von Mises, R. (1921), Uber die Wahrscheinlichkeit Seltener Ereignisse, Z. Angew. Math. Mech. 1, 121-124.

Wang, Y.H. (1989), From Poisson to compound Poisson ap- proximations, Math. Sci. 14, 38-49.

Wang, Y.H. (1992), Letter to the editor, Amer. Statist. 46, 77.

derivations of the compound poisson distribution and process

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