Return intervals of rare events in records with long-term persistence

Many natural records exhibit long-term correlations characterized by a power-law decay of the auto-correlation function, C(s) s−γ, with time lag s and correlation exponent 0<γ<1. We study, how the presence of such correlations affects the statistics of the return intervals rq for events above a certain threshold value q. We find that (a) the mean return interval Rq does not depend on γ, (b) the distribution of rq follows a stretched exponential, ln Pq(r) −(r/Rq)γ, and (c) the return intervals are also long-term correlated with the exponent γ, yielding clustering of both small and large return intervals. We provide indications that both the stretched exponential behaviour and the clustering of rare events can be seen in long temperature records.