Tail behavior of the empirical distribution function of convolutions

W. Albers, W.C.M. Kallenberg

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    Abstract

    Control charts based on convolutions require study of the tail behavior of the empirical distribution function of convolutions. It is well-known that this empirical distribution function at a fixed argument $x$ is asymptotically normal. The asymptotic normality is extended here to sequences $x_{n}$ tending to infinity at a suitable rate. At still larger $x_{n}$'s Poisson limiting distributions come in for the classical empirical distribution function. Surprisingly, this property does not generalize to its convolution counterpart, since for those $x_{n}$'s it is degenerate at $0$ with probability tending to 1. Exact inequalities for the tail behavior are presented as well.
    Original languageEnglish
    Place of PublicationEnschede
    PublisherUniversity of Twente
    Number of pages25
    Publication statusPublished - 2004

    Publication series

    NameMemorandum
    PublisherDepartment of Applied Mathematics, University of Twente
    No.1740
    ISSN (Print)0169-2690

    Keywords

    • MSC-62E20
    • MSC-62P30
    • MSC-62G30

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