Computable bounds of ${\ell}^2$-spectral gap for discrete Markov chains with band transition matrices

TL;DR

This paper derives computable bounds for the spectral gap of discrete Markov chains with band transition matrices, linking spectral properties to convergence rates and providing practical bounds via truncation methods.

Contribution

It introduces a new criterion based on the essential spectral radius for assessing spectral gaps in Markov chains with band matrices, connecting spectral and ergodic properties.

Findings

Spectral gap (SG2) holds if a specific limit supremum condition is met.

The essential spectral radius is bounded by a computable quantity .

Effective convergence bounds are obtainable through truncation.

Abstract

We analyse the $\ell^2(\pi)$-convergence rate of irreducible and aperiodic Markov chains with $N$-band transition probability matrix $P$ and with invariant distribution $\pi$. This analysis is heavily based on: first the study of the essential spectral radius $r\_{ess}(P\_{|\ell^2(\pi)})$ of $P\_{|\ell^2(\pi)}$ derived from Hennion's quasi-compactness criteria; second the connection between the Spectral Gap property (SG$\_2$) of $P$ on $\ell^2(\pi)$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\_2$) is shown to hold under the condition $ \alpha\_0 := \sum\_{{m}=-N}^N \limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\, 1 $ Moreover $r\_{ess}(P\_{|\ell^2(\pi)}) \leq \alpha\_0$. Effective bounds on the convergence rate can be provided from a truncation procedure.