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 rq 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 Cx(s)∼s−γ, with γ between 0 and 1. Here we study how the presence of long-term correlations changes the statistics of the return intervals rq. We find that (a) the mean return intervals Rq=〈rq〉 are independent of γ, (b) the distribution of the rq follows a stretched exponential, lnPq(r)∼−(r/Rq)γ, and (c) the return intervals are long-term correlated with an exponent γ′ close to γ.