Title: On the long range dependence property of fractional counting processes

Abstract

The fractional generalizations of the Poisson process has drawn the attention of many researchers since the last decade. Recent works on fractional extensions of the Poisson process, commonly known as the fractional Poisson processes, lead to some interesting connections between the areas of fractional calculus, stochastic subordination and renewal theory. The state probabilities of such processes are governed by the systems of fractional differential equations which display a slowly decreasing memory. It seems a characteristic feature of all real systems. Here, we discuss some recently introduced generalized counting processes and their fractional variants. Various fractional counting processes such as the fractional Poisson process and its mixed variants exhibit the longrange dependence property. It is proved by establishing an asymptotic result for the covariance of inverse stable subordinator.