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We study the concentration phenomenon for discrete-time random dynamicalsystems with an unbounded state space. We develop a heuristic approach towardsobtaining exponential concentration inequalities for dynamical systems using anentirely functional analytic framework. We also show that existence ofexponential-type Lyapunov function, compared to the purely deterministicsetting, not only implies stability but also exponential concentrationinequalities for sampling from the stationary distribution, via\emph{transport-entropy inequality} (T-E). These results have significantimpact in \emph{reinforcement learning} (RL) and \emph{controls}, leading toexponential concentration inequalities even for unbounded observables, whileneither assuming reversibility nor exact knowledge of random dynamical system(assumptions at heart of concentration inequalities in statistical mechanicsand Markov diffusion processes).

Transportation-Inequalities, Lyapunov Stability and Sampling for Dynamical Systems on Continuous State Space