Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 1, 28-41

ON A STUDY OF RENEWAL PROCESS CONNECTED WITH CERTAIN CONDITIONAL MOMENTS

By

WEN-JANG HUANG

And

JYH-CHERNG SU, National Sun Yat-sen University, Kaohsiung

SUMMARY. We prove that the inter-arrival times of a renewal ps {A(t), t >= 0}, with Sk being the kth arrival time, have a gamma distribution if for some integers n >= 2, r >=2, and 1<= k1 < k2 < ..<kr = n, E(Stki |A (t) = n), for every t >0 and i = 1, ., r-1. Under stronger conditions, characterizations of the Poisson process can be obtained. We also study the cases with negative order of conditional moments.

Key words and phrases. Characterization, conditional moment, Gamma distribution, martingale, Poison process, renewal process.

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