Time-changed Fractional Convoluted Skellam Process

Abstract

In this article, we introduce a convoluted version of Skellam process. We also give an alternate definition through its transition probabilities and discuss its distributional properties. Next, we consider the fractional version of convoluted Skellam process, which we call as fractional convoluted Skellam process (FCSP), and derive the probability generating function, mean, variance, covariance function, and establish its the long-range dependence property. Later, we introduce two time-changed variants of fractional convoluted Skellam process, in which we introduce time-change FCSP by Lévy subordinator and its first exit time, and call as TCFCSP-I and TCFCSP-II respectively. We also derive the law of iterated logarithm property for TCFCSP-I. Later we give the expression for the asymptotic behavior of moments of TCFCSP-II. MSC Classification: 60G22; 60G55