Statistical Properties of Chaotic Dynamical Systems: Non-Stationary Central Limit Theorems and Extreme Value Theory



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In this thesis, some statistical properties of two interesting problems are studied.

The first one is about non-stationary central limit theorems. We establish central limit theorems for a sequence of nested balls using martingale difference array technique, under certain conditions. It applies to various dynamical systems, including smooth expanding maps of the interval, Rychlik type maps, Gibbs-Markov maps, rational maps, and piecewise expanding maps in higher dimension.

And the second problem is about the Lorenz Systems. We study a family of Geometrical Lorenz Models, which have very similar properties to Lorenz Systems and are easier to study. Based on the model, we establish dynamical Borel Cantelli lemmas and convergence of rare event points processes to Poisson processes, which implies Extreme Value Theory.



Central limit theorem, Borel-Cantelli Lemma, Lorenz system, Extreme Value theory


Portions of this document appear in: Haydn, N., Nicol, M., Vaienti, S., and Zhang, L. (2013). Central limit theorems for the shrinking target problem. Journal of Statistical Physics, 153(5), 864-887.