# Quasi-stationary distributions for reducible absorbing Markov chains in discrete time

P.K. Pollett, Erik A. van Doorn

Research output: Book/ReportReportProfessional

## Abstract

We consider discrete-time Markov chains with one coffin state and a finite set $S$ of transient states, and are interested in the limiting behaviour of such a chain as time $n \to \infty,$ conditional on survival up to $n$. It is known that, when $S$ is irreducible, the limiting conditional distribution of the chain equals the (unique) quasi-stationary distribution of the chain, while the latter is the (unique) $\rho$-invariant distribution for the one-step transition probability matrix of the (sub)Markov chain on $S,$ $\rho$ being the Perron-Frobenius eigenvalue of this matrix. Addressing similar issues in a setting in which $S$ may be reducible, we identify all quasi-stationary distributions and obtain a necessary and sufficient condition for one of them to be the unique $\rho$-invariant distribution. We also reveal conditions under which the limiting conditional distribution equals the $\rho$-invariant distribution if it is unique. We conclude with some examples.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics 16 Published - Nov 2008

### Publication series

Name Memorandum / Department of Applied Mathematics University of Twente, Department of Applied Mathematics 10/1885 1874-4850 1874-4850

## Keywords

• limiting conditional distribution
• IR-65084
• Absorbing Markov chain
• MSC-60J10
• EWI-14022
• survival-time distribution
• MSC-15A18
• rho-invariant distribution
• METIS-252095