Return interval distribution of extreme events and long term memory

TL;DR

This paper derives an analytical expression for the distribution of return intervals between extreme events in long-range correlated systems, revealing a product of power law and stretched exponential forms, and explores its dependence on thresholds.

Contribution

It provides the first analytical formula for return interval distributions in long-range correlated time series, extending previous numerical findings.

Findings

Distribution is a product of power law and stretched exponential.

Analytical expression holds for large average return intervals.

Distribution depends on the threshold defining extreme events.

Abstract

The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.