Fractional non-homogeneous Poisson and P\'olya-Aeppli processes of order $k$ and beyond

TL;DR

This paper introduces fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k, deriving their governing equations, analyzing their covariance structures, and exploring long-range dependence properties.

Contribution

It presents novel fractional non-homogeneous processes of order k and characterizes their dynamics through non-local equations and dependence analysis.

Findings

Derived non-local governing equations for the processes

Analyzed covariance structures and long-range dependence

Extended the framework of fractional non-homogeneous processes

Abstract

We introduce two non-homogeneous processes: a fractional non-homogeneous Poisson process of order $k$ and and a fractional non-homogeneous P\'olya-Aeppli process of order $k$. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.