Group theoretic dimension of stationary symmetric \alpha-stable random fields

TL;DR

This paper explores the relationship between the growth rate of partial maxima in stationary symmetric -stable random fields and the underlying group theoretic properties of the transformations generating these fields, extending previous discrete-index results to continuous indices.

Contribution

It generalizes the connection between stable random fields and group theory from discrete to continuous parameter spaces, specifically ^d.

Findings

Established the link between growth rate and group properties for R^d-indexed fields.

Extended previous discrete results to continuous parameter cases.

Provides a framework for analyzing stable fields via group theory.

Abstract

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (2004a,b), where it was established, based on the seminal works of Rosi\'nski (1995,2000), that the growth rate is connected to the ergodic theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by Z^d in Roy and Samorodnitsky (2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to the fields indexed by R^d.