Limit Theorems for Non-stationary and Random Dynamical Systems

We study the limit behavior of non-stationary/random chaotic dynamical systems and prove a strong statistical limit theorem: (vector-valued) almost sure invariance principle for non-stationary dynamical systems and quenched (vector-valued) almost sure invariance principle for random dynamical systems. It is a matching of the trajectories of the dynamical system with a Brownian motion in such a way that the error is negligible in comparison with the Birkhoff sum. We develop a method called "reverse Gaussian approximation" and apply it to the classical block construction. We apply our results to the stationary chaotic systems which can be described by Young tower, and the (non)uniformly expanding non-stationary/random dynamical systems with intermittency or uniform spectral gap. Our results imply that the systems we study have many limit results that are satisfied by Brownian motion.