Time-changed generalized fractional Skellam process

Abstract

In this paper, we introduce and study two time-changed variants of the generalized fractional Skellam process. These are obtained by time-changing the generalized fractional Skellam process with an independent L\'evy subordinator with finite moments of any order and its inverse, respectively. We call the introduced processes the time-changed generalized fractional Skellam process-I (TCGFSP-I) and the time-changed generalized fractional Skellam process-II (TCGFSP-II), respectively. The probability generating function, moment generating function, moments, factorial moments, variance, covariance, {\it etc.}, are derived for the TCGFSP-I. We obtain a variant of the law of the iterated logarithm for it and establish its long-range dependence property. Several special cases of the TCGFSP-I are considered, and the associated system of governing differential equations is obtained. Later, some distributional properties and particular cases are discussed for the TCGFSP-II.