Ergodic Theory, Abelian Groups, and Point Processes Induced by Stable Random Fields

TL;DR

This paper investigates the weak convergence of point processes induced by stationary symmetric alpha-stable random fields, distinguishing between dissipative and conservative group actions, and introduces normalization techniques for the conservative case.

Contribution

It provides a novel analysis of point process limits for conservative actions using group theory and extreme value methods, extending known results from dissipative cases.

Findings

Weak convergence to cluster Poisson process in dissipative case

Identification of non-tightness and normalization needs in conservative case

Explicit computation of weak limits using group structure and extreme value theory

Abstract

We consider a point process sequence induced by a stationary symmetric alpha-stable (0 < alpha < 2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky (2004), that if the random field is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specific class of stable random fields generated by conservative actions whose effective dimensions can be computed using the structure theorem of finitely generated abelian groups. The corresponding point processes sequence is not tight and hence needs to be properly normalized in order to ensure weak convergence. This weak limit is computed using extreme value theory and some counting techniques.