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The fractional non-homogeneous Poisson process

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  • Leonenko, Nikolai
  • Scalas, Enrico
  • Trinh, Mailan

Abstract

We introduce a non-homogeneous fractional Poisson process by replacing the time variable in the fractional Poisson process of renewal type with an appropriate function of time. We characterize the resulting process by deriving its non-local governing equation. We further compute the first and second moments of the process. Eventually, we derive the distribution of arrival times. Constant reference is made to previous known results in the homogeneous case and to how they can be derived from the specialization of the non-homogeneous process.

Suggested Citation

  • Leonenko, Nikolai & Scalas, Enrico & Trinh, Mailan, 2017. "The fractional non-homogeneous Poisson process," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 147-156.
    • Handle: RePEc:eee:stapro:v:120:y:2017:i:c:p:147-156
    DOI: 10.1016/j.spl.2016.09.024
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    References listed on IDEAS

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    1. Wang, Xiao-Tian & Zhang, Shi-Ying & Fan, Shen, 2007. "Nonhomogeneous fractional Poisson processes," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 236-241.
    2. Nikolai Leonenko & Ely Merzbach, 2015. "Fractional Poisson Fields," Methodology and Computing in Applied Probability, Springer, vol. 17(1), pages 155-168, March.
    3. Wang, Xiao-Tian & Wen, Zhi-Xiong & Zhang, Shi-Ying, 2006. "Fractional Poisson process (II)," Chaos, Solitons & Fractals, Elsevier, vol. 28(1), pages 143-147.
    4. Nikos Yannaros, 1994. "Weibull renewal processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(4), pages 641-648, December.
    5. Mauro Politi & Taisei Kaizoji & Enrico Scalas, 2011. "Full characterization of the fractional Poisson process," Papers 1104.4234, arXiv.org.
    6. Molchanov, Ilya & Ralchenko, Kostiantyn, 2015. "Multifractional Poisson process, multistable subordinator and related limit theorems," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 95-101.
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    2. Davide Cocco & Massimiliano Giona, 2021. "Generalized Counting Processes in a Stochastic Environment," Mathematics, MDPI, vol. 9(20), pages 1-19, October.
    3. Kreer, Markus, 2022. "An elementary proof for dynamical scaling for certain fractional non-homogeneous Poisson processes," Statistics & Probability Letters, Elsevier, vol. 182(C).
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    5. Beghin, Luisa & Macci, Claudio & Ricciuti, Costantino, 2020. "Random time-change with inverses of multivariate subordinators: Governing equations and fractional dynamics," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6364-6387.

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