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