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Perturbation bounds for the stationary distributions of Markov chains. (English) Zbl 1264.60052

Summary: We are interested in an investigation of 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 norm-wise 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 mean first hitting times. Some \(V\)-normwise bounds are also derived based on the results by N. V. Kartashov [J. Sov. Math. 34, 1493–1498 (1986; Zbl 0594.60069)]. Moreover, we show how the bounds developed in this paper and one bound given in [E. Seneta, Adv. Appl. Probab. 20, No. 1, 228–230 (1988; Zbl 0645.60070)] can be extended to continuous-time Markov chains. Several examples are presented to illustrate our results or to compare our bounds with the known ones in the literature.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J27 Continuous-time Markov processes on discrete state spaces
15B51 Stochastic matrices
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