A family of random sup-measures with long-range dependence

TL;DR

This paper introduces a new family of self-similar, translation-invariant random sup-measures exhibiting long-range dependence, derived as limits of empirical measures from heavy-tailed processes with clustering of extremes.

Contribution

It develops a novel class of random sup-measures capturing long-range dependence and clustering, with a limit theorem for associated point-process convergence.

Findings

Characterizes long-range dependence via random closed sets

Establishes limit theorem for empirical sup-measures

Connects heavy-tailed processes with clustering of extremes

Abstract

A family of self-similar and translation-invariant random sup-measures with long-range dependence are investigated. They are shown to arise as the limit of the empirical random sup-measure of a stationary heavy-tailed process, inspired by an infinite urn scheme, where same values are repeated at several random locations. The random sup-measure reflects the long-range dependence nature of the original process, and in particular characterizes how locations of extremes appear as long-range clusters represented by random closed sets. A limit theorem for the corresponding point-process convergence is established.