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Computable bounds of an \(\ell^2\)-spectral gap for discrete Markov chains with band transition matrices. (English) Zbl 1351.60092

Summary: 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 two steps. First, the study of the essential spectral radius \(r_{\mathrm{ess}}(P_{|\ell^2(\pi)})\) of \(P_{|\ell^2(\pi)}\) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (\(\mathrm{SG}_{2}\)) of \(P\) on \(\ell^2(\pi)\) and the \(V\)-geometric ergodicity of \(P\). Specifically, the (\(\mathrm{SG}_{2}\)) is shown to hold under the condition \(\alpha_{0}:=\sum_{m=-N}^{N}\limsup_{i\to+\infty}(P(i,i+m)P^\ast(i+m,i)^{1/2}<1\). Moreover, \(r_{\mathrm{ess}}(P_{|\ell^2(\pi)}\leq\alpha_{0}\). Effective bounds on the convergence rate can be provided from a truncation procedure.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
47B07 Linear operators defined by compactness properties
60F99 Limit theorems in probability theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)