Limit theorems for non-local functionals of smooth Gaussian fields via quasi-association

TL;DR

This paper introduces a new method based on topological quasi-association to analyze non-local functionals of smooth Gaussian fields, establishing limit theorems and concentration bounds for these quantities.

Contribution

It develops a novel approach using quasi-association to derive limit theorems for non-local Gaussian field functionals, extending classical locality-based methods.

Findings

Established limit theorems for non-local functionals

Proved concentration bounds and a quantitative CLT

Derived the law of the iterated logarithm for these functionals

Abstract

Many classical objects of study related to the geometry/topology of smooth Gaussian fields (e.g., the volume, surface area or Euler characteristic of excursion sets) have a `locality' property which is crucial to their analysis. More recently, progress has been made in studying `non-local' quantities of such fields (e.g., the component/nodal count or the Betti numbers of excursion sets). In this work, we develop a new approach to analysing non-local functionals based on a form of topological quasi-association. We use this to establish a variety of limit theorems for approximately additive functionals on growing domains, including concentration bounds, a quantitative central limit theorem and the law of the iterated logarithm.