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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

Information

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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