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December 2015 Counting processes with Bernštein intertimes and random jumps
Enzo Orsingher, Bruno Toaldo
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J. Appl. Probab. 52(4): 1028-1044 (December 2015). DOI: 10.1239/jap/1450802751

Abstract

In this paper we consider point processesNf(t), t > 0, with independentincrements and integer-valued jumps whose distribution is expressed in terms ofBernštein functions f with Lévy measure ν. Weobtain the general expression of the probability generating functionsGf of Nf, the equationsgoverning the state probabilitiespkf of Nf,and their corresponding explicit forms. We also give the distribution of thefirst-passage times Tkf ofNf, and the related governing equation. We study indetail the cases of the fractional Poisson process, the relativistic Poissonprocess, and the gamma-Poisson process whose state probabilities have the formof a negative binomial. The distribution of the timesτjlj of jumps withheight lj(∑j=1rlj = k)under the condition N(t) = k for all these specialprocesses is investigated in detail.

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Enzo Orsingher. Bruno Toaldo. "Counting processes with Bernštein intertimes and random jumps." J. Appl. Probab. 52 (4) 1028 - 1044, December 2015. https://doi.org/10.1239/jap/1450802751

Information

Published: December 2015
First available in Project Euclid: 22 December 2015

zbMATH: 1334.60085
MathSciNet: MR3439170
Digital Object Identifier: 10.1239/jap/1450802751

Subjects:
Primary: 60G55
Secondary: 60G50

Rights: Copyright © 2015 Applied Probability Trust

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Vol.52 • No. 4 • December 2015
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