Point processes of the Poisson-Skellam family
Fabrizio Cinque, Enzo Orsingher
TL;DR
This work extends Skellam-type processes to a broad non-homogeneous setting by representing the generalized Skellam process as a finite linear combination of independent Poisson processes. It develops exact generating functions, moments, and decompositions, and establishes limit theorems and weak convergence results, including LLN/CLT-type behavior and hydrodynamic limits. The paper then introduces fractional and Bernstein-subordinator generalizations, deriving fractional integrals, time-changed representations, and Caputo-time fractional derivatives, with both non-homogeneous and homogeneous variants and connections to Lévy processes and compound Poisson structures. These contributions widen the modeling toolkit for bursty, state-dependent counting phenomena with flexible time and space fractional dynamics, enabling applications in fields requiring precise jump-size control, thinning, and fractional-time effects.
Abstract
We study a general non-homogeneous Skellam-type process with jumps of arbitrary fixed size. We express this process in terms of a linear combination of Poisson processes and study several properties, including the summation of independent processes of the same family, some possible decompositions (which present particularly interesting characteristics) and the limit behaviors. In the case of homogeneous rate functions, a compound Poisson representation and a discrete approximation are presented. Then, we study the fractional integral of the process as well as the iterated integral of the running average. Finally, we consider some time-changed versions related to Lévy subordinators, connected to the Bernstein functions, and to the inverses of stable subordinators.