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

Richard B. Paris, Vladimir Vinogradov

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    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 languageEnglish
    Pages (from-to)43-67
    Number of pages25
    JournalCommunications on Stochastic Analysis
    Issue number1
    Publication statusPublished - Mar 2015


    • Compound Poisson–geometric process
    • Asymptotic expansion
    • Aaverage process
    • Backward evolution
    • Branching particle system
    • Confluent hypergeometric function
    • Feller-diffusion approximation
    • Hougaard process
    • Large deviations
    • Leading error term
    • Local limit theorem
    • Long-time behavior
    • Mixed Poisson process
    • Modified Bessel function of the first kind
    • Poisson–exponential distribution
    • Poisson approximation
    • Pólya-Aeppli distribution
    • Pólya-Aeppli Lévy process
    • Short-term behavior
    • Weak convergence


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