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

• Published in: Physica A 2003-12, Vol.330 (1), p.1-7
• Main Authors: Bunde, Armin, F. Eichner, Jan, Havlin, Shlomo, Kantelhardt, Jan W.
• Format: Article
• Language: English
• Subjects: Fluctuation analysis; Long-term correlations; Rare events; Return periods; Time series
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2003.08.004

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)∼ 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 r q . We find that (a) the mean return intervals R q =〈 r q 〉 are independent of γ, (b) the distribution of the r q follows a stretched exponential, ln P q(r)∼−(r/R q) γ , and (c) the return intervals are long-term correlated with an exponent γ′ close to γ.