Some probabilistic properties of fractional point processes

In this article, the first hitting times of generalized Poisson processes Nf(t), related to Bernštein functions f are studied. For the space-fractional Poisson processes, Nα(t), t > 0 (corresponding to f = xα), the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed. The processes Nf(t) are time-changed Poisson processes N(Hf(t)) with subordinators Hf(t) and here we study N(∑nj=1 Hfj(t)) and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form N(gH,v(t)) where (gH,v(t)) are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process.