Generalised Extreme Statistics and Sum of Correlated Variables
Résumé
So-called "generalised extreme value distributions" have become popular fitting functions for fluctuations observed in many correlated systems. We show that generalised extreme value statistics --the statistics of the $k^\mathrm{th}$ 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 generalisations 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. I will conclude by presenting possible extension of this approach to extreme value statistics of correlated random variables.