# Skellam and Time-Changed Variants of the Generalized Fractional Counting Process Source: [Skellam and Time-Changed Variants of the Generalized Fractional Counting Process]() Collection index: [llms.txt]() Data updated: 2025-10-27T05:26:34.657Z Authors: K. K. Kataria, M. Khandakar Published: 2021-07-17T19:43:21.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()