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

We analyse the ${\mathrm{\ell }}^2(\pi )$-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 $r\mathrm{\_}ess(P\mathrm{\_}\mathrm{\mid }{\mathrm{\ell }}^2(\pi ))$ of $P\mathrm{\_}\mathrm{\mid }{\mathrm{\ell }}^2(\pi )$ derived from Hennion’s quasi-compactness criteria; second the connection between the spectral gap property (SG$\mathrm{\_}2$) of $P$ on ${\mathrm{\ell }}^2(\pi )$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\mathrm{\_}2$) 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 $r\mathrm{\_}ess(P\mathrm{\_}\mathrm{\mid }{\mathrm{\ell }}^2(\pi ))\le \alpha \mathrm{\_}0$. Simple conditions on asymptotic properties of $P$ and of its invariant probability distribution $\pi$ to ensure that $\alpha \mathrm{\_}0\text{textless}1$ are given. In particular this allows us to obtain estimates of the ${\mathrm{\ell }}^2(\pi )$-geometric convergence rate of random walks with bounded increments. The specific case of reversible $P$ is also addressed. Numerical bounds on the convergence rate can be provided via a truncation procedure. This is illustrated on the Metropolis-Hastings algorithm.