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Fractional discrete processes: compound and mixed Poisson representations. (English) Zbl 1294.26004

Summary: 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.

MSC:

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