Maxima of stationary systems of randomly time-changed L\'evy particles

TL;DR

This paper constructs new stationary max-infinitely divisible processes from time-changed Le9vy particles, revealing how random time changes influence dependence structures and extremal behavior beyond max-stability.

Contribution

It introduces a method to generate non-trivial stationary max-id processes via random time changes, linking their extremal behavior to max-stable Le9vy-Brown-Resnick processes.

Findings

Time change alters dependence structure of max-id processes.

Stationarity maintained through reconfiguration of initial particle positions.

Extremal behavior linked to max-stable Le9vy-Brown-Resnick processes in the MDA.

Abstract

We construct stationary max-infinitely divisible (max-id) processes from systems of randomly time-changed L\'evy particles. Classical examples without time change, such as the Brown-Resnick process, are, up to marginal transformations, max-stable. We show that random time change of the underlying particles alters the dependence structure of the max-id process and leads, in general, beyond the max-stable setting. At the same time, stationarity is preserved by a suitable reconfiguration of the starting points of the particle system. We then prove that the extremal behavior of the resulting max-id process is linked to an associated max-stable L\'evy-Brown-Resnick process through the max-domain of attraction (MDA). Thus, our work combines potential theory for Markov processes and extreme value theory to yield a large class of new, non-trivial stationary processes in the MDA of a given L\'evy-Brown-Resnick process. So far, specific examples of processes in the MDA are scarce in the literature.