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

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A. Maheshwari, P. Vellaisamy

Probability · Submitted 20 Jan 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property.

how it is stated · asserted 1.00 read the record →

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arXiv ID1601.05177 (version 1)
Pith IDIQXRUONW, Pith's permanent record of this paper and its review
CitationA. Maheshwari, P. Vellaisamy. "On the Long-range Dependence of Fractional Poisson and Negative Binomial Processes." arXiv:1601.05177 (2016). https://pith.science/paper/IQXRUONW
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,091 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.26 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.90 · compares with prior results 0.87 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. contribution 0.96 0.81 0.08 asserted 0.54
s1 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 $0<\beta<\frac{1}{3}$. result 0.88 0.93 0.23 asserted 1.00
s2 We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property. result 1.00 0.94 0.51 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper
Original 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.

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