On the long-range dependence of fractional Poisson and negative binomial processes

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 [2] and prove that the increments of the FPP has indeed the short-range dependence (SRD) property, when the fractional index β satisfies 0 < β < 1 3 . We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property.

Time-changed Poisson processes of order k

“…Definition 5. (see [23]) Let 0 ≤ s < t and s be fixed. Assume a stochastic process {X(t)} t≥0 has the correlation function Corr[X(s), X(t)] that satisfies…”

Correlation properties of continuous-time autoregressive processes delayed by the inverse of the stable subordinator

“…For brief discussion and other possible definitions of long-range dependence in various settings see Heyde & Yang (1997). Long-range dependence property in the form of Definition 5.1 was first used in Maheshwari & Vellaisamy (2016) and Maheshwari & Vellaisamy (2017). Moreover, equivalent form was used in Leonenko et al (2013a).…”

Limit theorems for the fractional nonhomogeneous Poisson process

“…Limit theorems for Cox processes have been studied by Grandell (1976) and Serfozo (1972a,b). Specifically for the FHPP, long-range dependence has been discussed in Maheshwari and Vellaisamy (2016), scaling limits have been derived in Meerschaert and Scheffler (2004) and discussed in the context of parameter estimation in Cahoy et al (2010).…”

“…Limit theorems for Cox processes have been studied by [27] and [48], [49]. Specifically for the FHPP, long-range dependence has been discussed in [38], scaling limits have been derived in [41] and discussed in the context of parameter estimation in [12].…”