Török, Andrew2020-06-022020-06-02May 20202020-05May 2020https://hdl.handle.net/10657/6611We 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.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).almost sure invariance principle, non-stationary dynamical system, random dynamical system, decay of correlation2020-06-02Thesisborn digital