On the Long-range Dependence of Fractional Poisson and Negative Binomial Processes

TL;DR

This paper investigates the dependence properties of fractional Poisson and negative binomial processes, correcting previous errors and establishing that FPP has short-range dependence for certain parameters, while FNBP exhibits long-range dependence.

Contribution

It corrects a prior proof error and clarifies the dependence properties of FPP and FNBP, showing FPP has SRD and FNBP has LRD.

Findings

FPP has SRD when 0<β<1/3

FNBP exhibits LRD

Increments of FNBP have SRD

Abstract

We study the long-range dependence (LRD) of the increments of the fractional Poisson process (FPP), the fractional negative binomial process (FNBP) and the increments of the FNBP. We first point out an error in the proof of Theorem 1 of Biard and Saussereau (2014) and prove that the increments of the FPP has indeed the short-range dependence (SRD) property, when the fractional index $\beta$ satisfies $0<\beta<\frac{1}{3}$. We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property.