Some Compound Fractional Poisson Processes

TL;DR

This paper introduces fractional variants of three compound Poisson processes, analyzes their statistical properties, and establishes their connection as limiting cases of a generalized counting process, expanding understanding of fractional stochastic models.

Contribution

It presents the first fractional versions of the Bell-Touchard, Poisson-logarithmic, and Pólya-Aeppli processes, detailing their properties and relationships to a generalized counting process.

Findings

Derived mean, variance, and covariance of the processes.

Established long-range dependence properties.

Provided multiple forms of the one-dimensional distributions.

Abstract

In this paper, we introduce and study fractional versions of three compound Poisson processes, namely, the Bell-Touchard process, the Poisson-logarithmic process and the generalized P\'olya-Aeppli process. It is shown that these processes are limiting cases of a recently introduced process by Di Crescenzo et al. (2016), namely, the generalized counting process. We obtain the mean, variance, covariance, long-range dependence property etc. for these processes. Also, we obtain several equivalent forms of the one-dimensional distribution of fractional versions of these processes.