Extreme value statistics and return intervals in long-range correlated uniform deviates

TL;DR

This paper analyzes extremal statistics and return intervals in long-range correlated uniform sequences, deriving limiting distributions and demonstrating their robustness across different stochastic processes.

Contribution

It provides analytical results for extremal distributions in correlated uniform sequences and extends findings to a broad class of stochastic processes.

Findings

Limiting distributions are unaffected by correlations but converge more slowly.

Distributions differ from the Weibull distribution for maxima.

Results generalize to various stochastic processes.

Abstract

We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider e.g., maximum relative to minimum are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself i.e., a Weibull distribution, reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form.