Stable Random Fields, Point Processes and Large Deviations

TL;DR

This paper studies the large deviation behavior of point processes derived from stationary symmetric stable random fields, revealing how underlying group structures influence these deviations and applying results to functionals like sums and maxima.

Contribution

It introduces a framework for analyzing large deviations in stable random fields based on ergodic and group theoretic structures, extending previous work to non-Gaussian settings.

Findings

Different large deviation behaviors depending on group structures

Application to partial sums and maxima of stable fields

Framework applicable to various functionals of stable fields

Abstract

We investigate the large deviation behaviour of a point process sequence based on a stationary symmetric stable non-Gaussian discrete-parameter random field using the framework of Hult and Samorodnitsky (2010). Depending on the ergodic theoretic and group theoretic structures of the underlying nonsingular group action, we observe different large deviation behaviours of this point process sequence. We use our results to study the large deviations of various functionals (e.g., partial sum, maxima, etc.) of stationary symmetric stable fields.