Nonsingular Group Actions and Stationary S alpha S Random Fields

TL;DR

This paper explores the connection between nonsingular group actions and the behavior of stationary symmetric alpha-stable random fields, extending existing structural results and analyzing how ergodic properties influence extreme values.

Contribution

It extends structure results of stationary S alpha S processes to fields and links ergodic properties of actions to field behavior.

Findings

Different ergodic types lead to distinct extreme behaviors.

Extensions of structure theorems from processes to fields.

Analysis of how nonsingular actions affect field properties.

Abstract

This paper deals with measurable stationary symmetric stable random fields indexed by R^d and their relationship with the ergodic theory of nonsingular R^d-actions. Based on the phenomenal work of Rosinski(2000), we establish extensions of some structure results of stationary S alpha S processes to S\alpha S fields. Depending on the ergodic theoretical nature of the underlying action, we observe different behaviors of the extremes of the field.