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The space-fractional Poisson process

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  • Orsingher, Enzo
  • Polito, Federico
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    Abstract

    In this paper, we introduce the space-fractional Poisson process whose state probabilities pkα(t), t≥0, α∈(0,1], are governed by the equations (d/dt)pkα(t)=−λα(1−B)αpkα(t), where (1−B)α is the fractional difference operator found in the time series analysis. We explicitly obtain the distributions pkα(t), the probability generating functions Gα(u,t), which are also expressed as distributions of the minimum of i.i.d. uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space–time-fractional Poisson process of which we give the explicit distribution.

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    Bibliographic Info

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

    Volume (Year): 82 (2012)
    Issue (Month): 4 ()
    Pages: 852-858

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    Handle: RePEc:eee:stapro:v:82:y:2012:i:4:p:852-858

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    Related research

    Keywords: Space-fractional Poisson process; Backward shift operator; Discrete stable distributions; Stable subordinator; Space–time fractional Poisson process;

    References

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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. Devroye, Luc, 1993. "A triptych of discrete distributions related to the stable law," Statistics & Probability Letters, Elsevier, vol. 18(5), pages 349-351, December.
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    Cited by:
    1. Beghin, Luisa & Macci, Claudio, 2013. "Large deviations for fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1193-1202.
    2. Orsingher, Enzo & Polito, Federico, 2013. "On the integral of fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1006-1017.

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