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Probability Theory and Related Fields

, Volume 147, Issue 3–4, pp 675–710 | Cite as

Hitting time statistics and extreme value theory

  • Ana Cristina Moreira Freitas
  • Jorge Milhazes FreitasEmail author
  • Mike Todd
Article

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.

Keywords

Return time statistics Extreme value theory Non-uniform hyperbolicity Interval maps 

Mathematics Subject Classification (2000)

37A50 37C40 60G10 60G70 37B20 37D25 37E05 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Ana Cristina Moreira Freitas
    • 1
  • Jorge Milhazes Freitas
    • 2
    Email author
  • Mike Todd
    • 2
  1. 1.Faculdade de Economia, Centro de MatemáticaUniversidade do PortoPortoPortugal
  2. 2.Centro de Matemática da Universidade do PortoPortoPortugal

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