{"paper":{"title":"Additional material on bounds of $\\ell^2$-spectral gap for discrete Markov chains with band transition matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"INSA Rennes), James Ledoux (IRMAR, Lo\\\"ic Herv\\'e (IRMAR","submitted_at":"2015-03-07T19:41:41Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02206","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}