Branching particle systems and compound Poisson processes related to Pólya-Aeppli distributions

Richard B. Paris; Vladimir Vinogradov

Branching particle systems and compound Poisson processes related to Pólya-Aeppli distributions

Richard B. Paris, Vladimir Vinogradov

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Abstract

We establish numerous new refined local limit theorems for a class of compound Poisson processes with Pólya-Aeppli marginals, and for a particular family of the branching particle systems which undergo critical binary branching and can be approximated by the backshifted Feller diffusion. To this end, we also derive new results for the families of Pólya–Aeppli and Poisson–exponential distributions. We relate a few of them to properties of certain special functions some of which were previously unknown.

Original language	English
Pages (from-to)	43-67
Number of pages	25
Journal	Communications on Stochastic Analysis
Volume	9
Issue number	1
Publication status	Published - Mar 2015

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Paris, R. B., & Vinogradov, V. (2015). Branching particle systems and compound Poisson processes related to Pólya-Aeppli distributions. Communications on Stochastic Analysis, 9(1), 43-67.

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abstract = "We establish numerous new refined local limit theorems for a class of compound Poisson processes with P{\'o}lya-Aeppli marginals, and for a particular family of the branching particle systems which undergo critical binary branching and can be approximated by the backshifted Feller diffusion. To this end, we also derive new results for the families of P{\'o}lya–Aeppli and Poisson–exponential distributions. We relate a few of them to properties of certain special functions some of which were previously unknown.",

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AB - We establish numerous new refined local limit theorems for a class of compound Poisson processes with Pólya-Aeppli marginals, and for a particular family of the branching particle systems which undergo critical binary branching and can be approximated by the backshifted Feller diffusion. To this end, we also derive new results for the families of Pólya–Aeppli and Poisson–exponential distributions. We relate a few of them to properties of certain special functions some of which were previously unknown.

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