Survival in a quasi-death process

Erik A. van Doorn, Philip K. Pollett

19 Citations (Scopus)

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 language Undefined 10.1016/j.laa.2008.04.004 776-791 14 Linear algebra and its applications 429 4 https://doi.org/10.1016/j.laa.2008.04.004 Published - 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.
@article{00486b5ca4a647719e1b73322d625f89,
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 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.",
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",
author = "{van Doorn}, {Erik A.} and Pollett, {Philip K.}",
note = "10.1016/j.laa.2008.04.004",
year = "2008",
doi = "10.1016/j.laa.2008.04.004",
language = "Undefined",
volume = "429",
pages = "776--791",
journal = "Linear algebra and its applications",
issn = "0024-3795",
publisher = "Elsevier",
number = "4",

}

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.

TY - JOUR

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.

KW - EWI-12808

KW - MSC-60J27

KW - M-matrix

KW - Quasi-stationary distribution

KW - survival-time distribution

KW - Absorbing Markov chain

KW - death process

KW - migration process

KW - METIS-250995

KW - IR-62322

KW - limiting conditional distribution

U2 - 10.1016/j.laa.2008.04.004

DO - 10.1016/j.laa.2008.04.004

M3 - Article

VL - 429

SP - 776

EP - 791

JO - Linear algebra and its applications

JF - Linear algebra and its applications

SN - 0024-3795

IS - 4

M1 - 10.1016/j.laa.2008.04.004

ER -