Journal of Applied Probability
- J. Appl. Probab.
- Volume 52, Number 4 (2015), 1028-1044.
Counting processes with Bernštein intertimes and random jumps
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 ν. 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 τjlj of jumps with height lj (∑j=1rlj = k) under the condition N(t) = k for all these special processes is investigated in detail.
J. Appl. Probab., Volume 52, Number 4 (2015), 1028-1044.
First available in Project Euclid: 22 December 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G55: Point processes
Secondary: 60G50: Sums of independent random variables; random walks
Orsingher, Enzo; Toaldo, Bruno. Counting processes with Bernštein intertimes and random jumps. J. Appl. Probab. 52 (2015), no. 4, 1028--1044. doi:10.1239/jap/1450802751. https://projecteuclid.org/euclid.jap/1450802751