Journal of Applied Probability

Fractional discrete processes: compound and mixed Poisson representations

Luisa Beghin and Claudio Macci

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We consider two fractional versions of a family of nonnegative integer-valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Pólya-Aeppli process, the Poisson inverse Gaussian process, and the negative binomial process. We also define and study some more general fractional versions with two fractional parameters.

Article information

J. Appl. Probab., Volume 51, Number 1 (2014), 19-36.

First available in Project Euclid: 25 March 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A33: Fractional derivatives and integrals 33E12: Mittag-Leffler functions and generalizations 60G22: Fractional processes, including fractional Brownian motion

Cox process doubly stochastic Poisson process negative binomial process Pólya-Aeppli process Poisson inverse Gaussian process


Beghin, Luisa; Macci, Claudio. Fractional discrete processes: compound and mixed Poisson representations. J. Appl. Probab. 51 (2014), no. 1, 19--36. doi:10.1239/jap/1395771411.

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