Extremal theory for long range dependent infinitely divisible processes

TL;DR

This paper develops new limit theorems for long memory infinitely divisible processes, revealing a family of non-classical limit distributions and random sup-measures governed by the process's memory and stable regenerative sets.

Contribution

It introduces a novel class of limit theorems involving phase transitions and new self-similar random sup-measures for long-range dependent processes.

Findings

Limits include interpolations between $ ext{Fréchet}$ and skewed $ ext{stable}$ distributions.

The limit objects are new stationary, self-similar random sup-measures.

Results extend previous work to broader parameter ranges.

Abstract

We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $\alpha$-Fr\'echet distribution and the skewed $\alpha$-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters $\alpha\in(0,\infty)$ and $\beta\in(0,1)$, with representations based on intersections of independent $\beta$-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index $-\alpha$. The intriguing structure of these random sup-measures is due to intersections of independent $\beta$-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as $\beta$ increases to one. The results in this paper extend substantially previous investigations where only $\alpha\in(0,2)$ and $\beta\in(0,1/2)$ have been considered.