QUASI-STATIONARITY OF DISCRETE-TIME MARKOV CHAINS

WITH DRIFT TO INFINITY

PAULINE COOLEN-SCHRIJNER^{⋆}, University of Durham

PHIL POLLETT^{⋆⋆}, The University of Queensland

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.

TRANSIENT MARKOV CHAINS, INVARIANT MEASURES, LIMITING CONDITIONAL DISTRIBU-TIONS, QUASI-STATIONARY DISTRIBUTIONS

AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J10 SECONDARY 60J15

⋆ Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UK. E-mail address: pauline.schrijner@durham.ac.uk ⋆⋆ Department of Mathematics, The University of Queensland, QLD 4072, Australia. E-mail address: pkp@maths.uq.edu.au The research of P. Pollett was supported by the Australia Research Council (Grant No. A69130032).

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.

Correspondence to first named author.