Hitting time statistics and extreme value theory

By A. C. Freitas, J. Freitas, Mike Todd. 2008

203 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/474d7306637ec5e8a0b24326457489daec1a9caf

How researchers see this paper

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In Mathematical Physics

1. Matthew Nicol
2. Mark Holland
3. Andrei Török
4. N. Haydn
5. Ian Melbourne
6. Alexander I. Bufetov
7. E. Rio

In Statistical and Nonlinear Physics

1. Mark Holland
2. Matthew Nicol
3. Andrei Török
4. N. Haydn
5. Ian Melbourne
6. Alexander I. Bufetov
7. E. Rio

Citations in the corpus

• Extreme value theory for non-uniformly expanding dynamical systems (2012, methodology): “Since |f ′ | is not uniformly bounded above, the conclusion does not immediately follow from the results of [4, 10] that we used to prove Theorem 3.”
• Entry and return times distribution (2013, background): “Recently Freitas, Freitas and Todd have established a relation between extremal value laws and entry times distribution first for absolutely continuous measures in [34] and then for general invariant measures in [35].”