A subgeometric estimate of the stability for time-homogeneous Markov chains.(Ukrainian, English)Zbl 1224.60030

Teor. Jmovirn. Mat. Stat. 81, 31-45 (2009); translation in Theory Probab. Math. Stat. 81, 35-50 (2010).
The author deals with two independent time-homogeneous Markov chains $$X$$ and $$X'$$ defined on a measurable space $$(E,\mathcal E)$$ with the transition probabilities $$P(x,A)$$ and $$P'(x,A)$$, respectively. In order to obtain an estimate of the stability, it is assumed that the chains are close to each other in the sense that $$P=(1-\varepsilon)Q+\varepsilon R$$ and $$P'=(1-\varepsilon)Q+\varepsilon R'$$, where $$Q$$ is the “common” part of two transition probabilities. The parameter $$\varepsilon$$ measures the closeness of two chains. It is shown that (as $$\varepsilon\to0$$) the difference of the transition probabilities after a sufficient number of steps tends to zero not faster than $$\varepsilon$$ does. Estimates for the stability of Markov chains are obtained with the help of the coupling method. The results are proved for both the uniform metric and for the nonuniform metric $$\| \cdot\| _v$$ which is introduced for a measure $$\mu$$ on $$(E,\mathcal E)$$ and a function $$v:E\to\mathbb R$$ by the relation $$\| \mu\| _v=\sup_{| g| \leq v}\left| \int_E\mu(dx)g(x)\right|$$. The proof of this result uses methods introduced by R. Douc, E. Moulines and P. Soulier [Bernoulli 13, No. 3, 831–848 (2007; Zbl 1131.60065)]. For more results on stability of Markov chains, see [N. V. Kartashov, Strong stable Markov chains. Utrecht: VSP. Kiev: TBiMC (1996; Zbl 0874.60082)].

MSC:

 60F05 Central limit and other weak theorems 60B10 Convergence of probability measures 60E15 Inequalities; stochastic orderings

Citations:

Zbl 1131.60065; Zbl 0874.60082
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