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Counting processes with Bernštein intertimes and random jumps
- Part of
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- Journal:
- Journal of Applied Probability / Volume 52 / Issue 4 / December 2015
- Published online by Cambridge University Press:
- 30 March 2016, pp. 1028-1044
- Print publication:
- December 2015
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- Article
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In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf , the equations governing the state probabilities pkf of Nf , and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf , and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times
of jumps with height lj (
) under the condition N(t) = k for all these special processes is investigated in detail.
of jumps with height 
) under the condition 