Computable bounds of ${\ell}^2$-spectral gap for discrete Markov chains with band transition matrices
| Autor: | Hervé, Loïc, Ledoux, James |
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| Rok vydání: | 2015 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: in Journal of Applied Probability, Applied Probability Trust, 2016. arXiv admin note: substantial text overlap with arXiv:1503.02206 |
| Databáze: | arXiv |
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