Survival in a quasi-death process

Erik A. van Doorn, Philip K. Pollett

    Research output: Book/ReportReportProfessional

    51 Downloads (Pure)

    Abstract

    We consider a Markov chain in continuous time with an absorbing coffin state and a finite set $S$ of transient states. When $S$ is irreducible the limiting distribution of the chain as $t \to\infty,$ conditional on survival up to time $t,$ is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which $S$ may be reducible, and obtain a complete solution if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has multiplicity one. The result is applied to pure death processes and, more generally, to quasi-death processes.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherUniversity of Twente, Department of Applied Mathematics
    Number of pages14
    Publication statusPublished - Jan 2007

    Publication series

    NameMemorandum / Department of Applied Mathematics
    PublisherUniversity of Twente, Department of Applied Mathematics
    No.1/1815
    ISSN (Print)1874-4850
    ISSN (Electronic)1874-4850

    Keywords

    • limiting conditional distribution
    • MSC-60J27
    • Absorbing Markov chain
    • Quasi-stationary distribution
    • EWI-8237
    • survival-time distribution
    • death process
    • METIS-241734
    • IR-66636

    Cite this

    van Doorn, E. A., & Pollett, P. K. (2007). Survival in a quasi-death process. (Memorandum / Department of Applied Mathematics; No. 1/1815). Enschede: University of Twente, Department of Applied Mathematics.
    van Doorn, Erik A. ; Pollett, Philip K. / Survival in a quasi-death process. Enschede : University of Twente, Department of Applied Mathematics, 2007. 14 p. (Memorandum / Department of Applied Mathematics; 1/1815).
    @book{7063b001d31045eebcf799d9c2e6eb37,
    title = "Survival in a quasi-death process",
    abstract = "We consider a Markov chain in continuous time with an absorbing coffin state and a finite set $S$ of transient states. When $S$ is irreducible the limiting distribution of the chain as $t \to\infty,$ conditional on survival up to time $t,$ is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which $S$ may be reducible, and obtain a complete solution if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has multiplicity one. The result is applied to pure death processes and, more generally, to quasi-death processes.",
    keywords = "limiting conditional distribution, MSC-60J27, Absorbing Markov chain, Quasi-stationary distribution, EWI-8237, survival-time distribution, death process, METIS-241734, IR-66636",
    author = "{van Doorn}, {Erik A.} and Pollett, {Philip K.}",
    year = "2007",
    month = "1",
    language = "Undefined",
    series = "Memorandum / Department of Applied Mathematics",
    publisher = "University of Twente, Department of Applied Mathematics",
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    van Doorn, EA & Pollett, PK 2007, Survival in a quasi-death process. Memorandum / Department of Applied Mathematics, no. 1/1815, University of Twente, Department of Applied Mathematics, Enschede.

    Survival in a quasi-death process. / van Doorn, Erik A.; Pollett, Philip K.

    Enschede : University of Twente, Department of Applied Mathematics, 2007. 14 p. (Memorandum / Department of Applied Mathematics; No. 1/1815).

    Research output: Book/ReportReportProfessional

    TY - BOOK

    T1 - Survival in a quasi-death process

    AU - van Doorn, Erik A.

    AU - Pollett, Philip K.

    PY - 2007/1

    Y1 - 2007/1

    N2 - We consider a Markov chain in continuous time with an absorbing coffin state and a finite set $S$ of transient states. When $S$ is irreducible the limiting distribution of the chain as $t \to\infty,$ conditional on survival up to time $t,$ is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which $S$ may be reducible, and obtain a complete solution if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has multiplicity one. The result is applied to pure death processes and, more generally, to quasi-death processes.

    AB - We consider a Markov chain in continuous time with an absorbing coffin state and a finite set $S$ of transient states. When $S$ is irreducible the limiting distribution of the chain as $t \to\infty,$ conditional on survival up to time $t,$ is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which $S$ may be reducible, and obtain a complete solution if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has multiplicity one. The result is applied to pure death processes and, more generally, to quasi-death processes.

    KW - limiting conditional distribution

    KW - MSC-60J27

    KW - Absorbing Markov chain

    KW - Quasi-stationary distribution

    KW - EWI-8237

    KW - survival-time distribution

    KW - death process

    KW - METIS-241734

    KW - IR-66636

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    T3 - Memorandum / Department of Applied Mathematics

    BT - Survival in a quasi-death process

    PB - University of Twente, Department of Applied Mathematics

    CY - Enschede

    ER -

    van Doorn EA, Pollett PK. Survival in a quasi-death process. Enschede: University of Twente, Department of Applied Mathematics, 2007. 14 p. (Memorandum / Department of Applied Mathematics; 1/1815).