TRANSPORT-INFORMATION INEQUALITIES FOR MARKOV CHAINS

• 成果类型: Article
• 署名作者: Wang, Neng-Yi; Wu, Liming
• 署名单位: Huazhong University of Science & Technology; Universite Clermont Auvergne (UCA); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1530
• 发表日期: 2020
• 页码: 1276-1320
• 关键词: asymptotic evaluation; process expectations; large deviations; random-variables; spectral gap; cost; probability; CURVATURE; time

摘要

This paper is the discrete time counterpart of the previous work in the continuous time case by Guillin, Leonard, the second named author and Yao [Probab. Theory Related Fields 144 (2009), 669-695]. We investigate the following transport-information TvI inequality: alpha(T-v(nu, mu)) <= I(nu vertical bar P, mu) for all probability measures nu on some metric space (X, d), where mu is an invariant and ergodic probability measure of some given transition kernel P (x, dy), T-v(nu, mu) is some transportation cost from nu to mu, I(nu vertical bar P, mu.) is the Donsker-Varadhan information of mu with respect to (P, mu) and alpha : [0, infinity) ->[0, infinity] is some left continuous increasing function. Using large deviation techniques, we show that TvI is equivalent to some concentration inequality for the occupation measure of the mu-reversible Markov chain (X-n)(n >= 0) with transition probability P (x, dy). Its relationships with the transport-entropy inequalities are discussed. Several easy-to-check sufficient conditions are provided for TvI. We show the usefulness and sharpness of our general results by a number of applications and examples. The main difficulty resides in the fact that the information I(nu vertical bar P, mu) has no closed expression, contrary to the continuous time or independent and identically distributed case.