Maximal coupling procedure and stability of discrete Markov chains. II

Abstract

Two discrete Markov chains whose one-step transition probabilities are close to each other in the uniform total variation norm or in the $V$-norm are considered. The problem of stability of the transition probabilities over an arbitrary number of steps is investigated. The main assumption is that either the uniform mixing or $V$-mixing condition holds. In particular, we prove that the uniform distance between the distributions of the chains after an arbitrary number of steps does not exceed $\varepsilon /(1-\rho )$, where $\varepsilon$ is the uniform distance between the transition matrices and where $\rho$ is the uniform mixing coefficient. A number of general examples are considered. The proofs are based on the maximal coupling procedure that maximizes the one-step coupling probabilities.