A fractional counting process and its connection with the Poisson process

TL;DR

This paper introduces a fractional counting process with jumps of various sizes, explores its properties through fractional differential equations, and connects it to classical fractional Poisson processes, revealing new distributional and asymptotic results.

Contribution

It develops a fractional counting process with multiple jump sizes, derives its distribution and moment generating function, and establishes its connection to fractional Poisson processes and subordinators.

Findings

First occurrence time matches classical fractional Poisson process

Distribution of first passage time expressed in integral form for k=2

Ratios of process powers over their means tend to 1 in probability

Abstract

We consider a fractional counting process with jumps of amplitude $1,2,\ldots,k$, with $k\in \mathbb{N}$, whose probabilities satisfy a suitable system of fractional difference-differential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When $k=2$ we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability.