On efficient estimates of the rate of convergence for Markov chains

TL;DR

This paper compares different methods for estimating the convergence rate of Markov chains in total variation, introducing a spectral method and modifications that can provide near-optimal bounds for both homogeneous and non-homogeneous chains.

Contribution

It introduces a spectral method and modifications to existing methods, demonstrating their potential to yield near-optimal convergence rate estimates for Markov chains.

Findings

Modified methods can provide near-optimal convergence estimates.

Spectral method offers an effective approach for convergence rate evaluation.

Theoretical results applicable to both homogeneous and non-homogeneous 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.