### 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 |
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Pages (from-to) | 191-204 |

Number of pages | 16 |

Journal | Markov processes and related fields |

Volume | 15 |

Issue number | 2 |

Publication status | Published - 2009 |

### Keywords

- MSC-60J10
- MSC-15A18
- survival-time distribution
- limiting conditional distribution
- Absorbing Markov chain
- EWI-15684
- METIS-264409
- IR-67801
- rho-invariant distribution

## Cite this

van Doorn, E. A., & Pollett, P. K. (2009). Quasi-stationary distributions for reducible absorbing Markov chains in discrete time.

*Markov processes and related fields*,*15*(2), 191-204.