Extremal clustering in non-stationary random sequences

TL;DR

This paper investigates the behavior of extreme values in non-stationary, identically distributed sequences with long-range dependence, revealing how extremal clustering influences their distribution and proposing estimators for clustering measures.

Contribution

It introduces a new framework for analyzing extremal clustering in non-stationary sequences and derives estimators for clustering measures under long-range dependence.

Findings

Limiting distribution depends on a parameter measuring extremal clustering.

Derived asymptotic distribution for the time between extreme events.

Constructed estimators for extremal clustering measures.

Abstract

It is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but identically distributed sequences of random variables subject to suitable long-range dependence restrictions. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence. Based on this new representation we derive the asymptotic distribution for the time between consecutive extreme observations and construct moment and likelihood-based estimators for measures of extremal clustering. We specialize our results to random sequences with periodic dependence structure.