Statistics of return intervals and extreme events in long-term correlated time series

We review our studies of the statistics of return intervals and extreme events (maxima) in long-term power-law correlated data sets characterized by correlation exponents gamma between 0 and 1 and different (Gaussian, exponential, power-law, and log-normal) distributions. We found that the long-term memory leads (i) to a stretched exponential distribution of the return intervals (Weibull distribution with an exponent equal to gamma), (ii) to clustering of both small and large return intervals, and (iii) to an anomalous behavior of the mean residual time to the next extreme event that increases with the elapsed time in a counterintuitive way. For maxima within time segments of fixed duration R we found that (i) the integrated distribution function converges to a Gumbel distribution for large R similar to uncorrelated signals, (ii) the speed of the convergence depends on both, the long-term correlations and the initial distribution of the values, (iii) the maxima series exhibit long-term correlations similar to those of the original data, and most notably (iv) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the previous maximum. Most of the effects revealed in artificial data can also be found in real hydro- and climatological data series.