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Space-Fractional Versions of the Negative Binomial and Polya-Type Processes

Author

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  • L. Beghin

    (Sapienza University of Rome)

  • P. Vellaisamy

    (Indian Institute of Technology Bombay)

Abstract

In this paper, we introduce a space fractional negative binomial process (SFNB) by time-changing the space fractional Poisson process by a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Lévy process and the corresponding Lévy measure is given. Extensions to the case of distributed order SFNB, where the fractional index follows a two-point distribution, are investigated in detail. The relationship with space fractional Polya-type processes is also discussed. Moreover, we define and study multivariate versions, which we obtain by time-changing a d-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications to population’s growth and epidemiology models are explored. Finally, we discuss algorithms for the simulation of the SFNB process.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:20:y:2018:i:2:d:10.1007_s11009-017-9561-8
    DOI: 10.1007/s11009-017-9561-8
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    References listed on IDEAS

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    1. Beghin, L., 2012. "Random-time processes governed by differential equations of fractional distributed order," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1314-1327.
    2. Jordi Valero & Josep Ginebra & Marta Pérez-Casany, 2012. "Extended Truncated Tweedie-Poisson Model," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 811-829, September.
    3. Orsingher, Enzo & Polito, Federico, 2012. "The space-fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 852-858.
    4. N. S. Upadhye & P. Vellaisamy, 2014. "Compound Poisson Approximation to Convolutions of Compound Negative Binomial Variables," Methodology and Computing in Applied Probability, Springer, vol. 16(4), pages 951-968, December.
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    Cited by:

    1. Yakubovich, Yuri, 2021. "A simple proof of the Lévy–Khintchine formula for subordinators," Statistics & Probability Letters, Elsevier, vol. 176(C).
    2. 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.
    3. Maheshwari, Aditya, 2023. "Tempered space fractional negative binomial process," Statistics & Probability Letters, Elsevier, vol. 196(C).

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