一般状态空间的马链蒙特卡罗算法在应用概率、Bayesian统计、机器学习、图像信号处理、经济金融等领域有重要的应用。在这些应用中，马链蒙特卡罗算法的非渐近定量性质对于评价算法的表现是极其重要的。分析这些非渐近定量性质的一个有力工具是测度集中理论。本项目将从两种角度来研究：(1)将研究重点提升到马氏链层面来；(2)结合统计模型研究两大类马链蒙特卡罗算法（Metropolis-Hastings 算法、Gibbs 算法）。为此本项目将要研究以下三个内容：构成马链蒙特卡罗算法的根本马氏链的集中不等式；Metropolis-Hastings 算法的集中不等式；基于统计模型的Gibbs 算法的集中不等式。

Markov chain Monte Carlo (MCMC) algorithms on general state space have important applications in applied probability, Bayesian statistics, machine learning, image and signal processing, economics and finance and so on. In these applications, it’s essential for evaluating the performance of algorithms to obtain non-asymptotic quantitative properties of MCMC. A powerful tool to analyze these non-asymptotic quantitative properties is the concentration of measure. This project is to investigate them from two perspectives: one is from the fundamental Markov chain, and the other is from Metropolis-Hastings algorithms and Gibbs samplings associated with statistical models. To this end, this project mainly contains the following contents: concentration inequalities for general underlying Markov chains (forming MCMC algorithms); concentration inequalities for Metropolis-Hastings algorithms; concentration inequalities for Gibbs samplings based on statistical models.