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 |
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Article number | 10.1016/j.laa.2008.04.004 |
Pages (from-to) | 776-791 |
Number of pages | 14 |
Journal | Linear algebra and its applications |
Volume | 429 |
Issue number | 4 |
DOIs | |
Publication status | 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