A New Shape of Extremal Clusters For Certain Stationary Semi-Exponential Processes With Moderate Long Range Dependence

TL;DR

This paper explores the shape of extremal clusters in stationary semi-exponential processes with moderate long-range dependence, revealing a new cluster structure supported by stable regenerative sets with diverse extremes.

Contribution

It introduces a novel shape of extremal clusters for semi-exponential tails, expanding understanding beyond power-like and lognormal-like tail behaviors.

Findings

Extremal clusters for semi-exponential tails form a new shape.

Stable regenerative sets support a variety of extremes.

The results extend the theory of extremal processes with long memory.

Abstract

Extremal clusters of stationary processes with long memory can be quite intricate. For certain stationary infinitely divisible processes with subexponential tails, including both power-like tails and certain lighter tails, e.g. lognormal-like tails, such clusters may take the shape of stable regenerative sets. In this paper we show that for semi-exponential tails, which are even lighter, a new shape of extremal clusters arises. In this case each stable regenerative set supports a random panoply of varying extremes.