Inequalities in theorems on consolidation of Markov chains. (English. Russian original) Zbl 0627.60023

Kartashov, N. V.

Theory Probab. Math. Stat. 34, 67-80 (1987); translation from Teor. Veroyatn. Mat. Stat. 34, 62-73 (1986).

Author’s abstract: Suppose that the transition kernel Q of a Markov chain X on the space (E,\({\mathcal F})\) is close in some operator topology to the kernel of a chain \(\bar X\) whose stationary projection carries \({\mathcal F}\)-measurable functions into \({\mathcal F}_ 0\)-measurable functions, where \({\mathcal F}_ 0\) is a certain sub-\(\sigma\)-algebra of \({\mathcal F}.\)
It is proved that the pre-limit distribution and the stationary distribution of X are approximated by the corresponding distributions of a chain Y with “averaged” transition kernel \(\Pi\) Q \(\Pi\), where \(\Pi\) is the stationary projection of \(\bar X.\)
Inequalities are given which estimate the accuracy of this approximation in a metric uniform in the time, and they are expressed in terms of the ergodic characteristics of the limit chains Y and \(\bar X\) (the norms of their generalized potentials).

Reviewer: J.C.Massé

MSC:

60E15	Inequalities; stochastic orderings
60J27	Continuous-time Markov processes on discrete state spaces
60J35	Transition functions, generators and resolvents

Keywords:

transition kernel; stationary projection; stationary distribution; ergodic characteristics