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

Golomozyj, V. V.

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)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

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

Keywords:

Markov chain; coupling method; transition probability; stability

Citations:

Zbl 1131.60065; Zbl 0874.60082