Asymptotic results for families of random variables having power series distributions

In this paper we consider suitable families of power series distributed random variables, and we study their asymptotic behavior in the fashion of large (and moderate) deviations. We also present two examples of fractional counting processes, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in cite{PoganyTomovski}, the second one is inspired by a model studied in cite{GarraOrsingherPolito}.