The Mittag-Leffler function in the thinning theory for renewal processes

TL;DR

This paper highlights the importance of the Mittag-Leffler distribution in the thinning theory of renewal processes with power-law waiting times, connecting classical and fractional Poisson processes.

Contribution

It revises and clarifies the original approach of Gnedenko and Kovalenko by explicitly incorporating the Mittag-Leffler function into renewal process thinning theory.

Findings

Mittag-Leffler distribution is central to renewal process thinning.

Revised approach improves understanding of fractional Poisson processes.

Connects classical renewal theory with fractional diffusion models.

Abstract

The main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in the theory of continuous random walk and consequently of fractional diffusion in a plenary lecture by the late Prof Gorenflo at a Seminar on Anomalous Transport held in Bad-Honnef in July 2006, published in a 2008 book. After recalling the basic theory of renewal processes including the standard and the fractional Poisson processes, here we have revised the original approach by Gnedenko and Kovalenko for convenience of the experts of stochastic processes who are not aware of the relevance of the Mittag-Leffler function