Survival in a quasi-death process

Erik A. van Doorn and Phil Pollett

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Abstract: 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 tends to 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 obtain a complete solution 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 applied to pure death processes and, more generally, to quasi-death processes.

Keywords: absorbing Markov chain, death process, limiting conditional distribution, quasi-stationary distribution, survival-time distribution

Acknowledgement: This worked was funded by the Australian Research Council.

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Last modified: 12th June 2007