Asymptotic results for families of power series distributions

TL;DR

This paper investigates the asymptotic behavior of families of power series distributed random variables, including examples related to fractional counting processes and their normalization via the Prabhakar function.

Contribution

It provides new asymptotic results for power series distributions and introduces examples involving fractional counting processes with novel normalization techniques.

Findings

Asymptotic behavior characterized for large and moderate deviations.

Examples demonstrate applications to fractional counting processes.

Normalizations expressed through the Prabhakar function.

Abstract

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}.