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Some characterizations of the Poisson process and geometric renewal process
- Part of
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- Journal:
- Journal of Applied Probability / Volume 30 / Issue 1 / March 1993
- Published online by Cambridge University Press:
- 14 July 2016, pp. 121-130
- Print publication:
- March 1993
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- Article
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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.
, with 
the sequence of interarrival times. We prove that, given a function 
holds for a single positive integer 
is a Poisson process. On the other hand, for a delayed renewal process 
with 
the residual life at 
is independent of 
is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function 