Quasistationarity of continuous-time Markov chains with positive drift

Coolen-Schrijner, P., Hart, A.G. and Pollett, P.K.

Abstract: We shall study continuous-time Markov chains on the nonnegative integers which are both irreducible and transient, and which exhibit discernible stationarity before drift to infinity ``sets in''. We will show how this `quasi' stationary behaviour can be modelled using a limiting conditional distribution: specifically, the limiting state probabilities conditional on not having left 0 for the last time. By way of a dual chain, obtained by killing the original process on last exit from 0, we invoke the theory of quasistationarity for absorbing Markov chains. We prove that the conditioned state probabilities of the original chain are equal to the state probabilities of its dual conditioned on non-absorption, thus allowing 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 quasistationary distribution in the usual sense, a similar statement is not possible for the original chain.

Keywords: Invariant measures, limiting conditional distributions, quasistationary 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 A. Hart and 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
• Andrew Hart, Discipline of Mathematics, The University of Queensland. agh@maths.uq.edu.au
• Phil Pollett, Discipline of Mathematics, The University of Queensland. pkp@maths.uq.edu.au

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Last modified: 21st March 1998