Tail behavior of the empirical distribution function of convolutions

Willem/Wim Albers, W.C.M. Kallenberg

    Research output: Book/ReportReportProfessional

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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.
    Albers, Willem/Wim ; Kallenberg, W.C.M. / Tail behavior of the empirical distribution function of convolutions. Enschede : University of Twente, Department of Applied Mathematics, 2004. 25 p. (Memorandum Faculty of Applied Mathematics; 1740).
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    title = "Tail behavior of the empirical distribution function of convolutions",
    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.",
    keywords = "MSC-62E20, EWI-3560, IR-65924, MSC-62P30, METIS-219690, MSC-62G30",
    author = "Willem/Wim Albers and W.C.M. Kallenberg",
    note = "Imported from MEMORANDA",
    year = "2004",
    language = "Undefined",
    series = "Memorandum Faculty of Applied Mathematics",
    publisher = "University of Twente, Department of Applied Mathematics",
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    Albers, WW & Kallenberg, WCM 2004, Tail behavior of the empirical distribution function of convolutions. Memorandum Faculty of Applied Mathematics, no. 1740, University of Twente, Department of Applied Mathematics, Enschede.

    Tail behavior of the empirical distribution function of convolutions. / Albers, Willem/Wim; Kallenberg, W.C.M.

    Enschede : University of Twente, Department of Applied Mathematics, 2004. 25 p. (Memorandum Faculty of Applied Mathematics; No. 1740).

    Research output: Book/ReportReportProfessional

    TY - BOOK

    T1 - Tail behavior of the empirical distribution function of convolutions

    AU - Albers, Willem/Wim

    AU - Kallenberg, W.C.M.

    N1 - Imported from MEMORANDA

    PY - 2004

    Y1 - 2004

    N2 - 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.

    AB - 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.

    KW - MSC-62E20

    KW - EWI-3560

    KW - IR-65924

    KW - MSC-62P30

    KW - METIS-219690

    KW - MSC-62G30

    M3 - Report

    T3 - Memorandum Faculty of Applied Mathematics

    BT - Tail behavior of the empirical distribution function of convolutions

    PB - University of Twente, Department of Applied Mathematics

    CY - Enschede

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    Albers WW, Kallenberg WCM. Tail behavior of the empirical distribution function of convolutions. Enschede: University of Twente, Department of Applied Mathematics, 2004. 25 p. (Memorandum Faculty of Applied Mathematics; 1740).