Journal
LINEAR ALGEBRA AND ITS APPLICATIONS
Volume 429, Issue 4, Pages 776-791Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.laa.2008.04.004
Keywords
absorbing Markov chain; death process; limiting conditional distribution; migration process; M-matrix; quasi-stationary distribution; survival-time distribution
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We consider a Markov chain in continuous time with one absorbing state and a finite set S of transient states. When S is irreducible the limiting distribution of the chain as t -> infinity, 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. (C) 2008 Elsevier Inc. All rights reserved.
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