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

Armin Bunde, Jan F. Eichner, Shlomo Havlin, Jan W. Kantelhardt

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87 Scopus citations


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 , 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 R q=〈rq〉 are independent of γ, (b) the distribution of the rq follows a stretched exponential, lnP q(r) ∼ -(r/Rq)γ, and (c) the return intervals are long-term correlated with an exponent γ′ close to γ.

Original languageEnglish
Pages (from-to)1-7
Number of pages7
JournalPhysica A: Statistical Mechanics and its Applications
Issue number1-2
StatePublished - 1 Dec 2003
EventRandomes and Complexity - Eilat, Israel
Duration: 5 Jan 20039 Jan 2003

Bibliographical note

Funding Information:
This work has been supported by the Bundesministerium für Bildung und Forschung and the Israel Science Foundation.


  • Fluctuation analysis
  • Long-term correlations
  • Rare events
  • Return periods
  • Time series


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