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Counting processes with Bernštein intertimes and random jumps

Part of: Stochastic processes

Published online by Cambridge University Press:  30 March 2016

Enzo Orsingher  and
Bruno Toaldo
Enzo Orsingher*
Affiliation:
Sapienza Università di Roma
Bruno Toaldo*
Affiliation:
Sapienza Università di Roma
*
∗Postal address: Dipartimento di Statistica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy.
∗Postal address: Dipartimento di Statistica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy.
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Abstract

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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.

Keywords

Lévy measureBernštein functionsubordinatornegative binomialbeta random variable

MSC classification

Primary: 60G55: Point processes
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1028 - 1044
Copyright
Copyright © Applied Probability Trust 2015 

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