Fractional discrete processes: compound and mixed Poisson representations

TL;DR

This paper introduces fractional versions of certain nonnegative integer valued processes, demonstrating their overdispersion and solving fractional Kolmogorov equations, with applications to well-known processes like Poisson and Negative Binomial.

Contribution

It develops fractional models for classical processes, providing new mathematical formulations and extending existing processes with fractional parameters.

Findings

Fractional processes solve fractional Kolmogorov equations.

Demonstrated overdispersion in fractional models.

Extended classical processes with fractional parameters.

Abstract

We consider two fractional versions of a family of nonnegative integer valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Polya-Aeppli, the Poisson Inverse Gaussian and the Negative Binomial. We also define and study some more general fractional versions with two fractional parameters.