- Volume 5, Number 3 (1999), 535-569.
Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications
Let be an aperiodic irreducible recurrent (not necessarily positive recurrent) Markov chain taking values on a countable unbounded subset of , its invariant measure and is a non-negative function defined on . We first find sufficient conditions under which (the corresponding result for the finiteness of was obtained by Tweedie). Then we obtain lower and upper bounds for the values of the invariant measure on the subsets of , that is, . These bounds are expressed in terms of first passage probabilities and the first exit time from . 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 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.
Bernoulli, Volume 5, Number 3 (1999), 535-569.
First available in Project Euclid: 27 February 2007
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Aspandiiarov, Sanjar; Iasnogorodski, Roudolf. Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications. Bernoulli 5 (1999), no. 3, 535--569. https://projecteuclid.org/euclid.bj/1172617202