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Some characterizations of the Poisson process and geometric renewal process

Part of: Distribution theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Wen-Jang Huang ,
Shun-Hwa Li  and
Jyh-Cherng Su
Shun-Hwa Li*
Affiliation:
National Sun Yat-sen University
*
Postal address for all authors: Institute of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, R.O.C.
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Abstract

Let γ t and δ t denote the residual life at t and current life at t, respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as holds for a single positive integer n, then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t, we find that for some fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t.

Keywords

EXPONENTIAL DISTRIBUTIONGEOMETRIC DISTRIBUTIONMIXED RENEWAL PROCESS

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 121 - 130
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

∗

Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 80–0208-MI 10–06.

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