Skellam and Time-Changed Variants of the Generalized Fractional Counting Process

TL;DR

This paper introduces a Skellam-type variant of the generalized counting process and its fractional version, analyzing their distributional properties, dependence structure, and effects of different time-changes.

Contribution

It develops the generalized Skellam process and its fractional variant, providing their distributional properties, integral representations, and dependence characteristics, extending the generalized counting process framework.

Findings

The GFSP's one-dimensional distributions are not infinitely divisible.

Integral representation for GFSP state probabilities is derived.

Long-range dependence property is established for GFSP.

Abstract

In this paper, we study a Skellam type variant of the generalized counting process (GCP), namely, the generalized Skellam process. Some of its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained. Its fractional version, namely, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator. It is observed that the GFSP is a Skellam type version of the generalized fractional counting process (GFCP) which is a fractional variant of the GCP. It is shown that the one-dimensional distributions of the GFSP are not infinitely divisible. An integral representation for its state probabilities is obtained. We establish its long-range dependence property by using its variance and covariance structure. Also, we consider two time-changed versions of the GFCP. These are obtained by time-changing the GFCP by an independent L\'evy subordinator and its inverse. Some particular cases of these time-changed processes are discussed by considering specific L\'evy subordinators.