# Survival in a quasi-death process

Erik A. van Doorn, Philip K. Pollett

21 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