Tail behavior of the empirical distribution function of convolutions

W. Albers, W.C.M. Kallenberg

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    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, Department of Applied Mathematics
    Number of pages25
    Publication statusPublished - 2004

    Publication series

    PublisherDepartment of Applied Mathematics, University of Twente
    ISSN (Print)0169-2690


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


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