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Fractional Discrete Processes: Compound and Mixed Poisson Representations

Part of: Real functions Stochastic processes Other special functions

Published online by Cambridge University Press:  30 January 2018

Luisa Beghin  and
Claudio Macci
Luisa Beghin*
Affiliation:
Sapienza Università di Roma
Claudio Macci*
Affiliation:
Università di Roma Tor Vergata
*
∗ Postal address: Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy. Email address: luisa.beghin@uniroma1.it
∗∗ Postal address: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Roma, Italy. Email address: macci@mat.uniroma2.it
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Abstract

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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.

Keywords

Cox processdoubly stochastic Poisson processnegative binomial processPólya-Aeppli processPoisson inverse Gaussian process

MSC classification

Primary: 26A33: Fractional derivatives and integrals 33E12: Mittag-Leffler functions and generalizations 60G22: Fractional processes, including fractional Brownian motion
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 19 - 36
Copyright
© Applied Probability Trust 

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