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) similar to s(-gamma), with time lag s and correlation exponent 0 < γ < 1. We study, bow the presence of such correlations affects the statistics of the return intervals r(q) for events above a certain threshold value q. We find that (a) the mean return interval R, does not depend on gamma, (b) the distribution of r(q) follows a stretched exponential, In P-q(r) similar to -(r/R-q)(gamma), and (c) the return intervals are also long-term correlated with the exponent gamma, 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. (C) 2004 Published by Elsevier B.V.