# Maxima of stable random fields, nonsingular actions and finitely generated abelian groups: A survey

## TL;DR

This survey explores how ergodic theory of nonsingular actions and the structure of finitely generated abelian groups influence the extreme values of stationary symmetric stable random fields indexed by multi-dimensional integer lattices.

## Contribution

It provides a comprehensive overview of the connections between ergodic theory, group actions, and stable random fields, highlighting recent developments and applications.

## Key findings

- Summarizes the structure theorem for finitely generated abelian groups.
- Explores the role of nonsingular group actions in extreme value analysis.
- Highlights recent references and developments in the field.

## Abstract

This is a self-contained introduction to the applications of ergodic theory of nonsingular (also known as quasi-invariant) group actions and the structure theorem for finitely generated abelian groups on the extreme values of stationary symmetric stable random fields indexed by $\mathbb{Z}^d$. It is based on a mini course given in the Eighth Lectures on Probability and Stochastic Processes (held in the Bangalore Centre of Indian Statistical Institute during December 6-10, 2013) except that a few recent references have been added in the concluding part. This article is a survey of existing work and the proofs are therefore skipped or briefly outlined.