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The space-fractional Poisson process

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  • Orsingher, Enzo
  • Polito, Federico

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.

Suggested Citation

  • Orsingher, Enzo & Polito, Federico, 2012. "The space-fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 852-858.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:4:p:852-858
    DOI: 10.1016/j.spl.2011.12.018
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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. 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. K. K. Kataria & M. Khandakar, 2021. "On the Long-Range Dependence of Mixed Fractional Poisson Process," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1607-1622, September.
    2. Ritik Soni & Ashok Kumar Pathak, 2024. "Generalized Iterated Poisson Process and Applications," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3216-3245, November.
    3. Beghin, Luisa & Macci, Claudio, 2022. "Non-central moderate deviations for compound fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 185(C).
    4. Tapiero, Charles S. & Vallois, Pierre, 2016. "Fractional randomness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 1161-1177.
    5. Ritik Soni & Ashok Kumar Pathak, 2024. "Generalized Fractional Risk Process," Methodology and Computing in Applied Probability, Springer, vol. 26(4), pages 1-17, December.
    6. K. K. Kataria & M. Khandakar, 2024. "Fractional Skellam Process of Order k," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1333-1356, June.
    7. 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.
    8. Roberto Garra & Enzo Orsingher & Federico Polito, 2018. "A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability," Mathematics, MDPI, vol. 6(1), pages 1-10, January.
    9. K. K. Kataria & M. Khandakar, 2022. "Generalized Fractional Counting Process," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2784-2805, December.
    10. Rohini Bhagwanrao Pote & Kuldeep Kumar Kataria, 2025. "On Erlang Queue with Multiple Arrivals and its Time-Changed Variant," Methodology and Computing in Applied Probability, Springer, vol. 27(3), pages 1-36, September.
    11. Beghin, Luisa & Macci, Claudio, 2017. "Asymptotic results for a multivariate version of the alternative fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 260-268.
    12. Davide Cocco & Massimiliano Giona, 2021. "Generalized Counting Processes in a Stochastic Environment," Mathematics, MDPI, vol. 9(20), pages 1-19, October.
    13. K. K. Kataria & P. Vellaisamy, 2019. "On Distributions of Certain State-Dependent Fractional Point Processes," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1554-1580, September.
    14. L. Beghin & P. Vellaisamy, 2018. "Space-Fractional Versions of the Negative Binomial and Polya-Type Processes," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 463-485, June.
    15. Neha Gupta & Arun Kumar, 2023. "Fractional Poisson Processes of Order k and Beyond," Journal of Theoretical Probability, Springer, vol. 36(4), pages 2165-2191, December.
    16. Orsingher, Enzo & Polito, Federico, 2013. "On the integral of fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1006-1017.
    17. Kataria, K.K. & Vellaisamy, P., 2017. "Saigo space–time fractional Poisson process via Adomian decomposition method," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 69-80.
    18. Shilpa Garg & Ashok Kumar Pathak & Aditya Maheshwari, 2025. "Tempered Space-Time Fractional Negative Binomial Process," Methodology and Computing in Applied Probability, Springer, vol. 27(2), pages 1-16, June.
    19. Laskin, Nick, 2024. "A new approach to constructing probability distributions of fractional counting processes," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
    20. Nikolai Leonenko & Ely Merzbach, 2015. "Fractional Poisson Fields," Methodology and Computing in Applied Probability, Springer, vol. 17(1), pages 155-168, March.
    21. Beghin, Luisa & Macci, Claudio, 2013. "Large deviations for fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1193-1202.
    22. Kataria, K.K. & Khandakar, M., 2022. "Extended eigenvalue–eigenvector method," Statistics & Probability Letters, Elsevier, vol. 183(C).
    23. Maheshwari, Aditya, 2023. "Tempered space fractional negative binomial process," Statistics & Probability Letters, Elsevier, vol. 196(C).

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