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4:00 pm Wednesday, February 25, 2009 Geometry-Analysis Seminar: Extreme Value Theory for Non-Uniformly Hyperbolic Dynamical Systemsby
Matthew Nicol (University of Houston) in HB 227- Given a stochastic process $(X_i)$ we may form a derived process of successive maxima by defining $M_n=\max \{X_1,X_2,...,X_n\}$. Extreme value theory is concerned with the limiting distribution of $(M_n)$ under linear scalings, i.e. are there constants $a_n$, $b_n$ such that $P(a_n M_n +b_n \le v) \rightarrow G(v)$ for some nondegenerate distribution $G$. It turns out that if $(X_i)$ is iid there are only three types of non-degenerate distribution possible (up to scale and location changes). This result is called the law of types. In a similar way if $(X_i)$ is iid with finite second moment then $P(\frac{1}{n} \sum_{i}^n X_i\le v)\rightarrow N(v)$ where $N$ is always a Gaussian distribution. We present recent results on extreme value theory for observations on deterministic dynamical systems, in particular for functions with multiple maxima and some degree of regularity on certain non-uniformly hyperbolic dynamical systems. We also obtain the distribution of extreme values for time-series of observations on discrete and continuous suspensions of certain non-uniformly hyperbolic dynamical systems via a general lifting theorem. The main result is that a broad class of observations on non-uniformly hyperbolic systems exhibit the same extreme value statistics as i.i.d processes with the same distribution function.
Submitted by damanik@rice.edu |