# On convergence rate for homogeneous Markov chains

## TL;DR

This paper investigates improved convergence rates for ergodic homogeneous Markov chains, extending previous results to more general settings without requiring a unique dominated measure, and compares new bounds with classical inequalities and eigenvalue methods.

## Contribution

It introduces generalized convergence bounds for Markov chains that do not rely on a unique dominated measure, broadening applicability.

## Key findings

- New convergence bounds outperform classical inequalities in certain cases.
- Comparison with Markov-Dobrushin inequality shows improved bounds.
- Examples demonstrate the effectiveness of the new bounds in finite state spaces.

## Abstract

Improved rates of convergence for ergodic homogeneous Markov chains are studied. In comparison to the earlier papers the setting is also generalised to the case without a unique dominated measure. Examples are provided where the new bound is compared with the classical Markov -- Dobrushin inequality and with the second eigenvalue of the transition matrix for finite state spaces.