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Hitting time statistics and extreme value theory
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  • Published: 29 April 2009

Hitting time statistics and extreme value theory

  • Ana Cristina Moreira Freitas1,
  • Jorge Milhazes Freitas2 &
  • Mike Todd2 

Probability Theory and Related Fields volume 147, pages 675–710 (2010)Cite this article

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  • 116 Citations

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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.

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Author information

Authors and Affiliations

  1. Faculdade de Economia, Centro de Matemática, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    Ana Cristina Moreira Freitas

  2. Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    Jorge Milhazes Freitas & Mike Todd

Authors
  1. Ana Cristina Moreira Freitas
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  2. Jorge Milhazes Freitas
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  3. Mike Todd
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Corresponding author

Correspondence to Jorge Milhazes Freitas.

Additional information

J. M. Freitas is partially supported by POCI/MAT/61237/2004 and M. Todd is supported by FCT grant SFRH/BPD/26521/2006. All three authors are supported by FCT through CMUP.

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Cite this article

Freitas, A.C.M., Freitas, J.M. & Todd, M. Hitting time statistics and extreme value theory. Probab. Theory Relat. Fields 147, 675–710 (2010). https://doi.org/10.1007/s00440-009-0221-y

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  • Received: 18 June 2008

  • Revised: 06 April 2009

  • Published: 29 April 2009

  • Issue Date: July 2010

  • DOI: https://doi.org/10.1007/s00440-009-0221-y

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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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