Asymptotic Behaviour of Level Sets of Needlet Random Fields

TL;DR

This paper proves a Central Limit Theorem for the boundary lengths of excursion sets of needlet random fields on the sphere, using advanced probabilistic techniques to analyze their asymptotic behavior at high energies.

Contribution

It establishes the asymptotic normality of boundary lengths of needlet random fields' excursion sets, employing Stein-Malliavin methods and Wiener chaos expansion.

Findings

Boundary lengths follow a Gaussian distribution asymptotically.

Variance of each chaotic component stabilizes after normalization.

No dominant chaotic component influences the limit.

Abstract

We consider sequences of needlet random fields defined as weighted averaged forms of spherical Gaussian eigenfunctions. Our main result is a Central Limit Theorem in the high energy setting, for the boundary lengths of their excursion sets. This result is based on Stein-Malliavin techniques and Wiener chaos expansion for nonlinear functionals of random fields. To this end, a careful analysis of the variances of each chaotic component of the boundary length is carried out, showing that they are asymptotically constant, after normalisation, for all terms of the expansion and no leading component arises.