The effect of long-term correlations on the return periods of rare events

The basic assumption of common extreme value statistics is that different events in a time record are uncorrelated. In this case, the return intervals r(q) of events above a given threshold size q are uncorrelated and follow the Poisson distribution. In recent years there is growing evidence that several hydro-meteorological and physiological records of interest (e.g. river flows, temperatures, heartbeat intervals) exhibit long-term correlations where the autocorrelation function decays as C-x(s) similar to s(-gamma), with gamma between 0 and 1. Here we study how the presence of long-term correlations changes the statistics of the return intervals r(q). We find that (a) the mean return intervals R-q = (r(q)) are independent of gamma, (b) the distribution of the r(q) follows a stretched exponential, In P-q(r) similar to -(r/R-q)(gamma), and (c) the return intervals are long-term correlated with an exponent gamma' close to gamma. (C) 2003 Elsevier B.V. All rights reserved.