Tail behavior of the empirical distribution function of convolutions

Willem/Wim 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 languageUndefined
    Place of PublicationEnschede
    PublisherUniversity of Twente, Department of Applied Mathematics
    Number of pages25
    Publication statusPublished - 2004

    Publication series

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

    Keywords

    • MSC-62E20
    • EWI-3560
    • IR-65924
    • MSC-62P30
    • METIS-219690
    • MSC-62G30

    Cite this

    Albers, WW., & Kallenberg, W. C. M. (2004). Tail behavior of the empirical distribution function of convolutions. (Memorandum Faculty of Applied Mathematics; No. 1740). Enschede: University of Twente, Department of Applied Mathematics.