Quasi-stationarity of discrete-time Markov chains with drift to infinity

Coolen-Schrijner, P. and Pollett, P.K.

Abstract: We consider a discrete-time Markov chain on the non-negative integers with drift to infinity and study the limiting behaviour of the state probabilities conditioned on not having left state 0 for the last time. Using a transformation, we obtain a dual Markov chain with an absorbing state such that absorption occurs with probability 1. We prove that the state probabilities of the original chain conditioned on not having left state 0 for the last time are equal to the state probabilities of its dual conditioned on non-absorption. This allows us to establish the simultaneous existence, and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always a quasi-stationary distribution in the usual sense, a similar statement is not possible for the original chain.

Keywords: Transient Markov chains, invariant measures, limiting conditional distributions, quasi-stationary distributions.

Acknowledgements: The research of P. Coolen-Schrijner was supported by TMR grant ERBFMBICT950067 of the European Community and was partly carried out while visiting The University of Queensland. The research of P. Pollett was supported by the Australia Research Council (Grant No. A69130032).

The authors:

• Pauline Coolen-Schrijner, Department of Mathematical Sciences, University of Durham. pauline.schrijner@durham.ac.uk
• Phil Pollett, Discipline of Mathematics, The University of Queensland. pkp@maths.uq.edu.au

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Last modified: 21st March 1998