# 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.

## Key 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.