We study the concentration phenomenon for discrete-time random dynamical systems with an unbounded state space. We develop a heuristic approach towards obtaining exponential concentration inequalities for dynamical systems using an entirely functional analytic framework. We also show that existence of exponential-type Lyapunov function, compared to the purely deterministic setting, not only implies stability but also exponential concentration inequalities for sampling from the stationary distribution, via \emph{transport-entropy inequality} (T-E). These results have significant impact in \emph{reinforcement learning} (RL) and \emph{controls}, leading to exponential concentration inequalities even for unbounded observables, while neither assuming reversibility nor exact knowledge of random dynamical system (assumptions at heart of concentration inequalities in statistical mechanics and Markov diffusion processes).

翻译：我们研究了具有不受约束状态空间的离散时间随机动态系统的集中现象。我们利用一个完全功能性分析框架,为动态系统获得指数性集中不平等制定了一种超常方法。我们还表明,与纯粹确定性环境相比,Lyapunov功能的指数性类型不仅意味着稳定性,而且还意味着固定分布(通过\emph{运输-有机物不平等}(T-E)进行抽样的指数性集中不平等。这些结果对\emph{强制学习}(RL)和\emph{控制}产生了重大影响,导致甚至对未受约束的可观测物的指数性集中不平等,同时既不假定可逆转性,也不确切了解随机动态系统(统计机学和Markov扩散过程中集中不平等的核心因素)。