Geometric ergodicity and the spectral gap of non-reversible Markov chains

• 成果类型: Article
• 署名作者: Kontoyiannis, I.; Meyn, S. P.
• 署名单位: Athens University of Economics & Business; University of Illinois System; University of Illinois Urbana-Champaign; University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0373-4
• 发表日期: 2012
• 页码: 327-339
• 关键词: CENTRAL-LIMIT-THEOREM; convergence-rates; bounds

摘要

We argue that the spectral theory of non-reversible Markov chains may often be more effectively cast within the framework of the naturally associated weighted-L (a) space , instead of the usual Hilbert space L (2) = L (2)(pi), where pi is the invariant measure of the chain. This observation is, in part, based on the following results. A discrete-time Markov chain with values in a general state space is geometrically ergodic if and only if its transition kernel admits a spectral gap in . If the chain is reversible, the same equivalence holds with L (2) in place of . In the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in but not in L (2). Moreover, if a chain admits a spectral gap in L (2), then for any there exists a Lyapunov function such that V (h) dominates h and the chain admits a spectral gap in . The relationship between the size of the spectral gap in or L (2), and the rate at which the chain converges to equilibrium is also briefly discussed.