Convoluted Fractional Poisson Process

TL;DR

This paper introduces the convoluted fractional Poisson process (CFPP), a new stochastic process derived from fractional differential equations, and explores its properties, special cases, and dependence structures.

Contribution

It presents the first detailed study of CFPP, including explicit formulas, statistical properties, and its relation to the convoluted Poisson process, a Lévy process.

Findings

CFPP's Laplace transform and distribution are derived.

CFPP exhibits short-range dependence, while CPP shows long-range dependence.

The process generalizes classical Poisson processes with fractional and convoluted features.

Abstract

In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. Some of its statistical properties such as probability generating function, moment generating function, moments etc. are obtained. A special case of CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a L\'evy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property.