%0 Book Section %1 statphys23_0097 %A Krug, J. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K change climatic dynamics events evolutionary extreme processes record statphys23 stochastic topic-3 %T Records in a changing world %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=97 %X A record is an entry in a sequence of random variables (RV's) that is larger or smaller than all previous entries. The mathematical theory of records is a subfield of extreme value statistics with a broad range of applications from climate research to evolutionary biology. After a quick review of the classic theory, which is largely restricted to sequences of independent and identically distributed (i.i.d.) RV's, this talk will present new results for sequences of independent RV's with distributions that broaden or sharpen with time. In particular, we show that when the width of the distribution grows as a power law in time $n$, the mean number of records is asymptotically of order $n$ for distributions with a power law tail (the Fréchet class of extremal value statistics), of order $(n)^2$ for distributions of exponential type (Gumbel class), and of order $n^1/(\nu+1)$ for distributions of bounded support (Weibull class), where the exponent $\nu$ describes the behaviour of the distribution at the upper (or lower) boundary. Simulations are presented which indicate that, in contrast to the i.i.d. case, the sequence of record breaking events is correlated in such a way that the variance of the number of records is asymptotically smaller than the mean.

@incollection{statphys23_0097, abstract = {A record is an entry in a sequence of random variables (RV's) that is larger or smaller than all previous entries. The mathematical theory of records is a subfield of extreme value statistics with a broad range of applications from climate research to evolutionary biology. After a quick review of the classic theory, which is largely restricted to sequences of independent and identically distributed (i.i.d.) RV's, this talk will present new results for sequences of independent RV's with distributions that broaden or sharpen with time. In particular, we show that when the width of the distribution grows as a power law in time $n$, the mean number of records is asymptotically of order $\ln n$ for distributions with a power law tail (the \textit{Fr\'echet class} of extremal value statistics), of order $(\ln n)^2$ for distributions of exponential type (\textit{Gumbel class}), and of order $n^{1/(\nu+1)}$ for distributions of bounded support (\textit{Weibull class}), where the exponent $\nu$ describes the behaviour of the distribution at the upper (or lower) boundary. Simulations are presented which indicate that, in contrast to the i.i.d. case, the sequence of record breaking events is correlated in such a way that the variance of the number of records is asymptotically smaller than the mean.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Krug, J.}, biburl = {https://www.bibsonomy.org/bibtex/2616a0cf8e5fe67324d9b98d7762a916c/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {e903288543817217f4e854b25eaa7529}, intrahash = {616a0cf8e5fe67324d9b98d7762a916c}, keywords = {change climatic dynamics events evolutionary extreme processes record statphys23 stochastic topic-3}, month = {9-13 July}, timestamp = {2007-06-20T10:16:11.000+0200}, title = {Records in a changing world}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=97}, year = 2007 }