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On the integral of fractional Poisson processes


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  • Orsingher, Enzo
  • Polito, Federico
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 83 (2013)
    Issue (Month): 4 ()
    Pages: 1006-1017

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    Handle: RePEc:eee:stapro:v:83:y:2013:i:4:p:1006-1017

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    Keywords: Mittag-Leffler generalized functions; Riemann–Liouville fractional integrals; Skellam distribution;


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    1. Mauro Politi & Taisei Kaizoji & Enrico Scalas, 2011. "Full characterization of the fractional Poisson process," Papers 1104.4234,
    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.


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