### 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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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 16 |

Publication status | Published - Nov 2008 |

### Publication series

Name | Memorandum / Department of Applied Mathematics |
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Publisher | University of Twente, Department of Applied Mathematics |

No. | 10/1885 |

ISSN (Print) | 1874-4850 |

ISSN (Electronic) | 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

## Cite this

Pollett, P. K., & van Doorn, E. A. (2008).

*Quasi-stationary distributions for reducible absorbing Markov chains in discrete time*. (Memorandum / Department of Applied Mathematics; No. 10/1885). Enschede: University of Twente, Department of Applied Mathematics.