An alternative formulation of the discrete-time fractional Poisson process

TL;DR

This paper proposes a new discrete-time fractional Poisson process based on renewal theory with Mittag-Leffler waiting times, highlighting differences from continuous-time models.

Contribution

It introduces a novel renewal-based discrete-time fractional Poisson process and derives its fundamental properties, contrasting it with existing models.

Findings

Derived the probability generating function of waiting times.

Obtained the exact probability distribution of event counts.

Showed the process differs from subordination-based models.

Abstract

This paper introduces a discrete-time fractional Poisson process defined as a renewal process, where the waiting times follow a discrete Mittag-Leffler distribution. We investigate its fundamental properties by explicitly deriving the probability generating function of the waiting times and the exact probability distribution of the event counts. Through this analysis, we reveal that, unlike its continuous-time counterpart, our renewal-based model is not mathematically equivalent to the process constructed via subordination using the Sibuya distribution.