Generalized extreme value statistics and sum of correlated variables Export

@article{citeulike:679729, abstract = {We show that generalized extreme value statistics\—the statistics of the k 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, Fr\'echet 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.}, author = {Bertin and Eric and Clusel and Maxime}, citeulike-article-id = {679729}, citeulike-linkout-0 = {http://dx.doi.org/10.1088/0305-4470/39/24/001}, citeulike-linkout-1 = {http://stacks.iop.org/0305-4470/39/7607}, day = {16}, doi = {10.1088/0305-4470/39/24/001}, issn = {0305-4470}, journal = {Journal of Physics A: Mathematical and General}, keywords = {extreme-value, gumbel, probability, statistics}, month = {June}, number = {24}, pages = {7607--7619}, posted-at = {2006-06-01 20:17:52}, priority = {4}, publisher = {Institute of Physics Publishing}, title = {Generalized extreme value statistics and sum of correlated variables}, url = {http://dx.doi.org/10.1088/0305-4470/39/24/001}, volume = {39}, year = {2006} }

TY - JOUR ID - citeulike:679729 L3 - citeulike-article-id:679729 N2 - We show that generalized extreme value statistics\—the statistics of the k 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, Fr\'echet 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. IS - 24 JF - Journal of Physics A: Mathematical and General SN - 0305-4470 EP - 7619 TI - Generalized extreme value statistics and sum of correlated variables PB - Institute of Physics Publishing VL - 39 SP - 7607 KW - extreme-value KW - gumbel KW - probability KW - statistics AU - Bertin AU - Eric AU - Clusel AU - Maxime PY - 2006/06/16/ UR - http://dx.doi.org/10.1088/0305-4470/39/24/001 ER -