COUNTING PROCESSES WITH BERNSTEIN INTERTIMES AND RANDOM JUMPS

In this paper we consider point processes N-f (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure nu. We obtain the general expression of the probability generating functions G(f) of N-f, the equations governing the state probabilities p(k)(f) of N-f, and their corresponding explicit forms. We also give the distribution of the first-passage times T-k(f) of N-f, 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 whole state probabilities have the form of a negative binomial. The distribution of the times tau(lj)(j) of jumps with height l(j) (Sigma(r)(j=1) l(j) = k) under the condition N(t) = k for all these special processes is investigated in detail.