On relations between extreme value statistics, extreme random matrices and Peak-Over-Threshold method

11/09/2017

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by   Jacek Grela, et al.

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Using the thinning method, we explain the link between classical Fisher-Tippett-Gnedenko classification of extreme events and their free analogue obtained by Ben Arous and Voiculescu in the context of free probability calculus. In particular, we present explicit examples of large random matrix ensembles, realizing free Weibull, free Fréchet and free Gumbel limiting laws, respectively. We also explain, why these free laws are identical to Balkema-de Haan-Pickands limiting distribution for exceedances, i.e. why they have the form of generalized Pareto distributions and derive a simple exponential relation between classical and free extreme laws.