Statistics of return intervals in long-term correlated records

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

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117 Scopus citations


We consider long-term correlated data with several distribution densities (Gaussian, exponential, power law, and log normal) and various correlation exponents γ (0<γ<1), and study the statistics of the return intervals rj between events above some threshold q. We show that irrespective of the distribution, the return intervals are long-term correlated in the same way as the original record, but with additional uncorrelated noise. Due to this noise, the correlations are difficult to observe by the detrended fluctuation analysis (which exhibits a crossover behavior) but show up very clearly in the autocorrelation function. The distribution Pq (r) of the return intervals is characterized at large scales by a stretched exponential with exponent γ, and at short scales by a power law with exponent γ-1. We discuss in detail the occurrence of finite-size effects for large threshold values for all considered distributions. We show that finite-size effects are most pronounced in exponentially distributed data sets where they can even mask the stretched exponential behavior in records of up to 106 data points. Finally, in order to quantify the clustering of extreme events due to the long-term correlations in the return intervals, we study the conditional distribution function and the related moments. We find that they show pronounced memory effects, irrespective of the distribution of the original data.

Original languageEnglish
Article number011128
JournalPhysical Review E
Issue number1
StatePublished - 2007


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