CHARACTERIZATIONS OF THE POISSON PROCESS AS A RENEWAL PROCESS VIA TWO CONDITIONAL MOMENTS

SHUN-HWA LI, WEN-JANG 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_{k}^{r} | A(t) = n) = at^{r} and E(S_{k}^{s} | 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.