Regularly varying random fields and analyses of extremal clusters

Abstract

The notion of regular variation is a common way to model the extreme events. While many researches have been conducted to study the regularly varying time series, its higher dimensional extension has not been well studied. In this work, we first study the extremes of multivariate regularly varying random fields using the tail field and the spectral field, notions that extend the tail and spectral processes of Basrak and Segers (2009). We discuss several properties and the limit theorems for the related point processes. The spatial context requires multiple notions of extremal index, and the tail and spectral fields are applied to clarify these notions and other aspects of extremal clusters. An important application of the techniques we develop is to the Brown-Resnick random fields. Next, we talk about the shapes of the extremal clusters. We first define the most likely clusters and provide some examples as an illustration. We discuss how the choice of the threshold level affects extremal clusters geographically. A special type of regularly varying random field we focus on is the shot noise model. Also, we introduce the duration of an extreme event and develop its limiting distribution. Extremal clusters are not only helpful in understanding extremes, but it also helps in the task such as understanding the underlying process. We provide a new way to estimate the parameters of AR(p) process with regularly varying noises using extremal clusters. Combined with a clustering (classification) algorithm, we are able to detect the regime changes and group different extremal clusters.