Full characterization of the fractional Poisson process

TL;DR

This paper provides a complete analytical characterization of the fractional Poisson process, including explicit formulas for its finite-dimensional distributions, enhancing understanding of its properties and differences from the classical Poisson process.

Contribution

The paper derives exact formulas for the finite-dimensional distributions of the fractional Poisson process, fully characterizing its non-Markovian and non-Lévy nature.

Findings

Derived explicit formulas for finite-dimensional distributions

Validated analytical results with Monte Carlo simulations

Clarified the process's non-stationary and non-Markovian properties

Abstract

The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poisson process, the fractional Poisson process does not have stationary and independent increments. It is not a L\'evy process and it is not a Markov process. In this letter, we present formulae for its finite-dimensional distribution functions, fully characterizing the process. These exact analytical results are compared to Monte Carlo simulations.