Maxima of stationary systems of randomly time-changed Lévy particles

Abstract

In this work, we consider maxima of systems of randomly time-changed Lévy particles. We give a general construction to obtain infinite-dimensional classes $\{Z^{\alpha }\}$ of stationary max-infinitely divisible (max-id) processes. These classes are indexed by admissible mass functions $\alpha$, which induce state-dependent time changes of the underlying Lévy particles. This gives a generalization of the well-known (Lévy--)Brown--Resnick process $Z^1$. In contrast to $\alpha \equiv 1$, the variability of non-constant mass functions $\alpha$ changes the dependence structure of the max-id process and goes beyond the max-stable setting while preserving stationarity. We then explore the extent of the so-called max-domain of attraction (MDA) of a given (Lévy--)Brown--Resnick process $Z^1$, by studying convergence of rescaled maxima of independent copies of $Z^{\alpha }$ to $Z^1$. Thus, our work combines potential theory for Markov processes and extreme value theory to yield a novel, infinite-dimensional, and interpretable class $\{Z^{\alpha }\}$ of stationary processes in the MDA of a given (Lévy--)Brown--Resnick process $Z^1$. So far, results on the extent of such domains have been scarce in the literature.

26 pages