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June 1999 Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications
Sanjar Aspandiiarov, Roudolf Iasnogorodski
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Bernoulli 5(3): 535-569 (June 1999).

Abstract

Let { Z n ,n0} be an aperiodic irreducible recurrent (not necessarily positive recurrent) Markov chain taking values on a countable unbounded subset S of R d , π () its invariant measure and f is a non-negative function defined on S . We first find sufficient conditions under which S f(z)π(dz)= (the corresponding result for the finiteness of S f(z)π(dz) was obtained by Tweedie). Then we obtain lower and upper bounds for the values of the invariant measure π on the subsets B of S , that is, π (B) . These bounds are expressed in terms of first passage probabilities and the first exit time from B . We also show how to estimate the latter quantities using sub- or supermartingale techniques. The results are finally illustrated for driftless reflected random walks in + 2 and for Markov chains on non-negative reals with asymptotically small drift of Lamperti type. In both cases we obtain very precise information on the asymptotic behaviour of their stationary measures.

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Sanjar Aspandiiarov. Roudolf Iasnogorodski. "Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications." Bernoulli 5 (3) 535 - 569, June 1999.

Information

Published: June 1999
First available in Project Euclid: 27 February 2007

zbMATH: 0948.60068
MathSciNet: MR1693596

Keywords: occupation time , recurrent Markov chain , reflected random walk , stationary measure , submartingale , supermartingale

Rights: Copyright © 1999 Bernoulli Society for Mathematical Statistics and Probability

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Vol.5 • No. 3 • June 1999
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