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Scientific article
English

Generalized extreme value statistics and sum of correlated variables

Published inJournal of physics. A, mathematical and general, vol. 39, no. 24, p. 7607-7619
Publication date2006-06-16
First online date2006-05-31
Abstract

We show that generalized extreme value statistics—the statistics of the kth largest value among a large set of random variables—can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Fréchet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumbel distributions with non-integer index k in the statistics of global observables. This is one of the very few known generalizations of the central limit theorem to non-independent random variables. Finally, in the context of a simple physical model, we relate the index k to the ratio of the correlation length to the system size, which remains finite in strongly correlated systems.

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Citation (ISO format)
BERTIN, Eric, CLUSEL, Maxime. Generalized extreme value statistics and sum of correlated variables. In: Journal of physics. A, mathematical and general, 2006, vol. 39, n° 24, p. 7607–7619. doi: 10.1088/0305-4470/39/24/001
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ISSN of the journal0305-4470
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