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Speed of convergence for laws of rare events and escape rates

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  • Freitas, Ana Cristina Moreira
  • Freitas, Jorge Milhazes
  • Todd, Mike

Abstract

We obtain error terms on the rate of convergence to Extreme Value Laws, and to the asymptotic Hitting Time Statistics, for a general class of weakly dependent stochastic processes. The dependence of the error terms on the ‘time’ and ‘length’ scales is very explicit. Specialising to data derived from a class of dynamical systems we find even more detailed error terms, one application of which is to consider escape rates through small holes in these systems.

Suggested Citation

  • Freitas, Ana Cristina Moreira & Freitas, Jorge Milhazes & Todd, Mike, 2015. "Speed of convergence for laws of rare events and escape rates," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1653-1687.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:4:p:1653-1687
    DOI: 10.1016/j.spa.2014.11.011
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    References listed on IDEAS

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    1. W. J. Hall & Jon A. Wellner, 1979. "The rate of convergence in law of the maximum of an exponential sample," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 33(3), pages 151-154, September.
    2. Freitas, Ana Cristina Moreira & Freitas, Jorge Milhazes, 2008. "On the link between dependence and independence in extreme value theory for dynamical systems," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1088-1093, July.
    3. Abadi, Miguel & Saussol, Benoit, 2011. "Hitting and returning to rare events for all alpha-mixing processes," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 314-323, February.
    4. Aldous, David J., 1982. "Markov chains with almost exponential hitting times," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 305-310, September.
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    Cited by:

    1. Freitas, Ana Cristina Moreira & Freitas, Jorge Milhazes & Todd, Mike & Vaienti, Sandro, 2016. "Rare events for the Manneville–Pomeau map," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3463-3479.
    2. Holland, M.P. & Nicol, M. & Török, A., 2016. "Almost sure convergence of maxima for chaotic dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 126(10), pages 3145-3170.

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