Non-homogeneous space-time fractional Poisson processes

TL;DR

This paper introduces the non-homogeneous space-time fractional Poisson process (NSTFPP), extending existing fractional Poisson processes, and analyzes its properties, limit theorems, dependence structure, and simulations.

Contribution

It provides the first detailed study of the NSTFPP, including its pmf, generating function, differential equation, and long-range dependence properties.

Findings

Derived the pmf and generating function of NSTFPP

Established limit theorems and LRD properties for NSTFPP

Presented simulated sample paths of the process

Abstract

The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto in \cite{sfpp}, is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NSTFPP). We compute their {\it pmf} and generating function and investigate the associated differential equation. The limit theorems and the law of iterated logarithm for the NSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NTFPP. We investigate the limit theorem and the LRD property for the fractional non-homogeneous Poisson process (FNPP), studied by Leonenko et. al. (2016). Finally, we present some simulated sample paths of the NSTFPP process.