Survival in a quasi-death process

Erik A. van Doorn, Philip K. Pollett

    Research output: Contribution to journalArticleAcademicpeer-review

    19 Citations (Scopus)
    28 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 show that it remains valid if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has geometric (but not, necessarily, algebraic) multiplicity one. The result is then applied to pure death processes and, more generally, to quasi-death processes. We also show that the result holds true even when the geometric multiplicity is larger than one, provided the irreducible subsets of $S$ satisfy an accessibility constraint. A key role in the analysis is played by some classic results on $M$-matrices.
    Original languageUndefined
    Article number10.1016/j.laa.2008.04.004
    Pages (from-to)776-791
    Number of pages14
    JournalLinear algebra and its applications
    Volume429
    Issue number4
    DOIs
    Publication statusPublished - 2008

    Keywords

    • EWI-12808
    • MSC-60J27
    • M-matrix
    • Quasi-stationary distribution
    • survival-time distribution
    • Absorbing Markov chain
    • death process
    • migration process
    • METIS-250995
    • IR-62322
    • limiting conditional distribution

    Cite this

    van Doorn, Erik A. ; Pollett, Philip K. / Survival in a quasi-death process. In: Linear algebra and its applications. 2008 ; Vol. 429, No. 4. pp. 776-791.
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    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 show that it remains valid if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has geometric (but not, necessarily, algebraic) multiplicity one. The result is then applied to pure death processes and, more generally, to quasi-death processes. We also show that the result holds true even when the geometric multiplicity is larger than one, provided the irreducible subsets of $S$ satisfy an accessibility constraint. A key role in the analysis is played by some classic results on $M$-matrices.",
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    van Doorn, EA & Pollett, PK 2008, 'Survival in a quasi-death process', Linear algebra and its applications, vol. 429, no. 4, 10.1016/j.laa.2008.04.004, pp. 776-791. https://doi.org/10.1016/j.laa.2008.04.004

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

    In: Linear algebra and its applications, Vol. 429, No. 4, 10.1016/j.laa.2008.04.004, 2008, p. 776-791.

    Research output: Contribution to journalArticleAcademicpeer-review

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    T1 - Survival in a quasi-death process

    AU - van Doorn, Erik A.

    AU - Pollett, Philip K.

    N1 - 10.1016/j.laa.2008.04.004

    PY - 2008

    Y1 - 2008

    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 show that it remains valid if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has geometric (but not, necessarily, algebraic) multiplicity one. The result is then applied to pure death processes and, more generally, to quasi-death processes. We also show that the result holds true even when the geometric multiplicity is larger than one, provided the irreducible subsets of $S$ satisfy an accessibility constraint. A key role in the analysis is played by some classic results on $M$-matrices.

    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 show that it remains valid if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has geometric (but not, necessarily, algebraic) multiplicity one. The result is then applied to pure death processes and, more generally, to quasi-death processes. We also show that the result holds true even when the geometric multiplicity is larger than one, provided the irreducible subsets of $S$ satisfy an accessibility constraint. A key role in the analysis is played by some classic results on $M$-matrices.

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    KW - Absorbing Markov chain

    KW - death process

    KW - migration process

    KW - METIS-250995

    KW - IR-62322

    KW - limiting conditional distribution

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