Showing papers by "Jorge Milhazes Freitas published in 2008"
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TL;DR: In this article, the authors show that a multimodal map with an invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example).
Abstract: We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to non-uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example). We also give applications of our theory to higher dimensional examples, for which we also obtain classical extreme value laws and exponential hitting time statistics (for balls). We extend these ideas to the subsequent returns to asymptotically small sets, linking the Poisson statistics of both processes.
133 citations
TL;DR: The authors reformulated the standard conditions that allow to reduce the study of extremes for dependent sequences to the classical extreme value theory and weaken the mixing type condition in such a way that, in the context of dynamical systems, it should follow from decay of correlations.
100 citations
TL;DR: In this article, the authors considered the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks-Carleson parameter.
Abstract: We consider the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks–Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes X0,X1,… , given by Xn=fan, for every integer n≥0, where each random variable Xn is distributed according to the unique absolutely continuous, invariant probability of fa. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of Mn=max {X0,…,Xn−1} is the same as that which would apply if the sequence X0,X1,… was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of Mn is of type III (Weibull).
43 citations