Survival in a quasi-death process

Erik A. van Doorn, Philip K. Pollett

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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.
    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

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