Hitting Time Statistics and Extreme Value Theory

Details Hitting Time Statistics and Extreme Value Theory Hitting Time Statistics and Extreme Value Theory

2008-04-17

arxiv.org/abs/0804.2887v2

Probab. Theory Related Fields, 147, no. 3, 2010, 675-710

Mathematics

Dynamical Systems

35 pages, 1 figure, higher dimensional example added

Scientific paper

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.

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