A Lower Bound on the Mixing Time of Uniformly Ergodic Markov Chains in Terms of the Spectral Radius

TL;DR

This paper establishes a theoretical lower bound on the mixing time of uniformly ergodic, reversible Markov chains based on the spectral radius, extending known finite state space results to general state spaces.

Contribution

Provides the first proof of a spectral radius-based lower bound on mixing time for general state space Markov chains, previously known only in finite cases.

Findings

Bound applies to uniformly ergodic, reversible chains

Extends finite state space results to general state spaces

Supports the spectral radius as a key parameter for mixing time

Abstract

We give a bound on the mixing time of a uniformly ergodic, reversible Markov chain in terms of the spectral radius of the transition operator. This bound has been established previously in finite state spaces, and is widely believed to hold in general state spaces, but a proof has not been provided to our knowledge.