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On efficient estimates of the rate of convergence for Markov chains

Alexander Veretennikov

Abstract

The paper presents efficient approaches for evaluating convergence rate in total variation for finite and general linear Markov chains. The motivation for studying convergence rate in this metric is its usefulness in various limit theorems. For homogeneous Markov chains the goal is to compare several different methods: (1) the second eigenvalue for the transition matrix method (the method no. 1), (2) the method based on Markov -- Dobrushin's ergodic coefficient, and the new spectral method developed in earlier works, as well as modifications of they both by iterations (the ``other methods''). We answer the question whether or not the ``other methods'' may provide the optimal or close to optimal convergence rate in the case of homogeneous Markov chains. The answer turns out to be positive for appropriate modifications of both ``other methods''. The analogues of these ``other methods'' for the non-homogeneous Markov chains are also presented. The work is theoretical. However, the methods of computing efficient bounds of convergence rates may be in demand in various applied areas.

On efficient estimates of the rate of convergence for Markov chains

Abstract

The paper presents efficient approaches for evaluating convergence rate in total variation for finite and general linear Markov chains. The motivation for studying convergence rate in this metric is its usefulness in various limit theorems. For homogeneous Markov chains the goal is to compare several different methods: (1) the second eigenvalue for the transition matrix method (the method no. 1), (2) the method based on Markov -- Dobrushin's ergodic coefficient, and the new spectral method developed in earlier works, as well as modifications of they both by iterations (the ``other methods''). We answer the question whether or not the ``other methods'' may provide the optimal or close to optimal convergence rate in the case of homogeneous Markov chains. The answer turns out to be positive for appropriate modifications of both ``other methods''. The analogues of these ``other methods'' for the non-homogeneous Markov chains are also presented. The work is theoretical. However, the methods of computing efficient bounds of convergence rates may be in demand in various applied areas.
Paper Structure (5 sections, 15 theorems, 96 equations)

This paper contains 5 sections, 15 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Homogeneous coupling & operators $V, V^{(m)}$
  3. Gantmacher's, MD and MD-$m$ methods
  4. Operators $V^{(m)}$ and $\hat{V}^{(m)}$ and their spectral radii
  5. Non-homogeneous case

Key Result

Proposition 1

Let there exist $m\ge 1$ such that Then the process $(X_n)$ is ergodic, i.e., there exists a limiting probability measure $\mu$, which is stationary and for every $n$, and for any $m\ge 1$

Theorems & Definitions (23)

  • Proposition 1: VerVer, proposition 1
  • Proposition 2: Gant, corollary of the formula (13.96) in ch. XIII, § 7
  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Remark 2
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • ...and 13 more