Computable 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_{ess}(P_{|\ell^2(\pi)}) \leq \alpha_0$. Effective bounds on the convergence rate can be provided from a truncation procedure.