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CHARACTERIZATIONS OF THE POISSON PROCESS AS

A RENEWAL PROCESS VIA TWO CONDITIONAL MOMENTS

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SHUN-HWA LI, WEN-J
ANG HUANG AND MONG-NA LO HUANG

*Institute of Applied Mathematics, National Sun Yat-sen University,*

Kaohsiung, Taiwan, 80424, R.O.C.
(Received July, 27, 1992; revised August 2, 1993)

**Abstract.**
Given two independent positive random
variables, under some minor conditions, it is known that from
*E*(*X*^{r} | *X* + *Y*) = *a*(*X*+*Y*)^{r} and *E*(*X*^{s} | *X*+*Y*) = *b*(*X*+*Y*)^{s}, for
certain pairs of *r* and *s*, where *a* and *b* are two
constants, we can characterize *X* and *Y* to have gamma
distributions. Inspired by this, in this article we will
characterize the Poisson process among the class of renewal
processes via two conditional moments. More precisely, let
{*A*(*t*), t __>__ 0} be a renewal process, with {*S*_{k}, *k* __>__ 1}
the sequence of arrival times, and *F* the common distribution
function of the inter-arrival times. We prove that for some fixed
*n* and *k*, *k* __<__ *n*, if *E*(*S*^{r}_{k} | *A*(*t*) = *n*) = *at*^{r} and
*E*(*S*^{s}_{k} | *A*(*t*) = *n*) = *bt*^{s}, for certain pairs of *r* and *s*,
where *a* and *b* are independent of *t*, then {*A*(*t*), *t* __>__ 0}
has to be a Poisson process. We also give some corresponding
results about characterizing *F* to be geometric when *F* is
discrete.

*Key words and phrases*:
Characterization,
exponential distribution, gamma distribution, geometric
distribution, Poisson process, renewal process.

**Source**
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