Perturbation Bounds for the Stationary Distributions of Markov Chains

Abstract

In this paper, we are interested in investigating the perturbation bounds for the stationary distributions for discrete-time or continuous-time Markov chains on a countable state space. For discrete-time Markov chains, two new normwise bounds are obtained. The first bound is rather easy to obtain since the needed condition, equivalent to uniform ergodicity, is imposed on the transition matrix directly. The second bound, which holds for a general (possibly periodic) Markov chain, involves finding a drift function. This drift function is closely related to the mean first hitting times. Some $V$-normwise bounds are also derived based on the results in [N. V. Kartashov, J. Soviet Math., 34 (1986), pp. 1493--1498]. Moreover, we show how the bounds developed in this paper and one bound given in [E. Seneta, Adv. Appl. Probab., 20 (1988), pp. 228--230] can be extended to continuous-time Markov chains. Several examples are shown to illustrate our results or to compare our bounds with the known ones in the literature.