Let [math] be an aperiodic irreducible recurrent (not necessarily positive recurrent) Markov chain taking values on a countable unbounded subset [math] of [math] , [math] its invariant measure and [math] is a non-negative function defined on [math] . We first find sufficient conditions under which [math] (the corresponding result for the finiteness of [math] was obtained by Tweedie). Then we obtain lower and upper bounds for the values of the invariant measure [math] on the subsets [math] of [math] , that is, [math] . These bounds are expressed in terms of first passage probabilities and the first exit time from [math] . 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 [math] 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.
Publié le : 1999-06-14
Classification:
occupation time,
recurrent Markov chain,
reflected random walk,
stationary measure,
submartingale,
supermartingale
@article{1172617202,
author = {Aspandiiarov, Sanjar and Iasnogorodski, Roudolf},
title = {Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications},
journal = {Bernoulli},
volume = {5},
number = {6},
year = {1999},
pages = { 535-569},
language = {en},
url = {http://dml.mathdoc.fr/item/1172617202}
}
Aspandiiarov, Sanjar; Iasnogorodski, Roudolf. Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications. Bernoulli, Tome 5 (1999) no. 6, pp. 535-569. http://gdmltest.u-ga.fr/item/1172617202/