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Asymptotic results for a multivariate version of the alternative fractional Poisson process

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  • Beghin, Luisa
  • Macci, Claudio

Abstract

A multivariate fractional Poisson process was recently defined in Beghin and Macci (2016) by considering a common independent random time change for a finite dimensional vector of independent (non-fractional) Poisson processes; moreover it was proved that, for each fixed t≥0, it has a suitable multinomial conditional distribution of the components given their sum. In this paper we consider another multivariate process {M̲ν(t)=(M1ν(t),…,Mmν(t)):t≥0} with the same conditional distributions of the components given their sums, and different marginal distributions of the sums; more precisely we assume that the one-dimensional marginal distributions of the process ∑i=1mMiν(t):t≥0 coincide with the ones of the alternative fractional (univariate) Poisson process in Beghin and Macci (2013). We present large deviation results for {M̲ν(t)=(M1ν(t),…,Mmν(t)):t≥0}, and this generalizes the result in Beghin and Macci (2013) concerning the univariate case. We also study moderate deviations and we present some statistical applications concerning the estimation of the fractional parameter ν.

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  • 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.
    • Handle: RePEc:eee:stapro:v:129:y:2017:i:c:p:260-268
    DOI: 10.1016/j.spl.2017.06.009
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    References listed on IDEAS

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    1. Kumar, A. & Nane, Erkan & Vellaisamy, P., 2011. "Time-changed Poisson processes," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1899-1910.
    2. Balakrishnan, N. & Kozubowski, Tomasz J., 2008. "A class of weighted Poisson processes," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2346-2352, October.
    3. Orsingher, Enzo & Polito, Federico, 2012. "The space-fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 852-858.
    4. Bryc, Wlodzimierz, 1993. "A remark on the connection between the large deviation principle and the central limit theorem," Statistics & Probability Letters, Elsevier, vol. 18(4), pages 253-256, November.
    5. Beghin, Luisa & Macci, Claudio, 2013. "Large deviations for fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1193-1202.
    6. Mauro Politi & Taisei Kaizoji & Enrico Scalas, 2011. "Full characterization of the fractional Poisson process," Papers 1104.4234, arXiv.org.
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    Cited by:

    1. Hainaut, Donatien, 2022. "Multivariate claim processes with rough intensities: Properties and estimation," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 269-287.
    2. 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.
    3. 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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