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Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution

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Abstract

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. (Phys. Rev. Lett. 97(21): 210602, 2006) have found analytical results.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading, RG6 6AX, UK

    Davide Faranda & Valerio Lucarini

  2. Department of Meteorology, Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Reading, RG6 6AX, UK

    Valerio Lucarini

  3. Department of Physics, INFN-Bologna, University of Bologna, Via Irnerio 46, Bologna, 40126, Italy

    Giorgio Turchetti

  4. UMR-6207, Centre de Physique Théorique, CNRS, Universités d’Aix-Marseille I, II, Université du Sud Toulon-Var and FRUMAM (Fédération de Recherche des Unités de Mathématiques de Marseille), CPT, Luminy, Case 907, 13288, Marseille Cedex 09, France

    Sandro Vaienti

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  1. Davide Faranda
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  2. Valerio Lucarini
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Correspondence to Davide Faranda.

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Faranda, D., Lucarini, V., Turchetti, G. et al. Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution. J Stat Phys 145, 1156–1180 (2011). https://doi.org/10.1007/s10955-011-0234-7

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