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

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Pollett, P.K. and van Doorn, E.A. (2008) Quasi-stationary distributions for reducible absorbing Markov chains in discrete time. [Report]

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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.
Item Type:Report
Faculty:
Electrical Engineering, Mathematics and Computer Science (EEMCS)
Research Group:
Link to this item:http://purl.utwente.nl/publications/65084
Official URL:http://www.math.utwente.nl/publications
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