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On the integral of fractional Poisson processes

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  • Orsingher, Enzo
  • Polito, Federico

Abstract

In this paper, we consider the Riemann–Liouville fractional integral Nα,ν(t)=1Γ(α)∫0t(t−s)α−1Nν(s)ds, where Nν(t), t≥0, is a fractional Poisson process of order ν∈(0,1], and α>0. We give the explicit bivariate distribution Pr{Nν(s)=k,Nν(t)=r}, for t≥s, r≥k, the mean ENα,ν(t) and the variance VarNα,ν(t). We study the process Nα,1(t) for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on N1,1(t) are presented in the last section where also distributional properties of the integrated Poisson process (including the representation as random sums) is derived. The integral of powers of the Poisson process is examined and its connections with generalized harmonic numbers are discussed.

Suggested Citation

  • Orsingher, Enzo & Polito, Federico, 2013. "On the integral of fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1006-1017.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:4:p:1006-1017
    DOI: 10.1016/j.spl.2012.12.016
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    References listed on IDEAS

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    1. Mauro Politi & Taisei Kaizoji & Enrico Scalas, 2011. "Full characterization of the fractional Poisson process," Papers 1104.4234, arXiv.org.
    2. Orsingher, Enzo & Polito, Federico, 2012. "The space-fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 852-858.
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    Cited by:

    1. Kreer, Markus & Kızılersü, Ayşe & Thomas, Anthony W., 2014. "Fractional Poisson processes and their representation by infinite systems of ordinary differential equations," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 27-32.
    2. A. Maheshwari & P. Vellaisamy, 2019. "Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1278-1305, September.
    3. Vishwakarma, P. & Kataria, K.K., 2024. "On integrals of birth–death processes at random time," Statistics & Probability Letters, Elsevier, vol. 214(C).

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