Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields

TL;DR

This paper investigates the asymptotic Gaussian behavior of spherical random fields, linking harmonic analysis, quantum angular momentum coupling, and applications to cosmology, especially the Cosmic Microwave Background.

Contribution

It introduces new conditions for Gaussianity of subordinated spherical fields and connects these results to random walks on SO(3), extending previous work on Abelian groups.

Findings

Established conditions for asymptotic Gaussianity in high-frequency limits.

Linked random walk behavior on SO(3) to spherical field analysis.

Provided estimates involving Clebsch-Gordan coefficients for convergence proofs.

Abstract

We study the weak convergence (in the high-frequency limit) of the frequency components associated with Gaussian-subordinated, spherical and isotropic random fields. In particular, we provide conditions for asymptotic Gaussianity and we establish a new connection with random walks on the the dual of SO(3), which mirrors analogous results previously established for fields defined on Abelian groups (see Marinucci and Peccati (2007)). Our work is motivated by applications to cosmological data analysis, and specifically by the probabilistic modelling and the statistical analysis of the Cosmic Microwave Background radiation, which is currently at the frontier of physical research. To obtain our main results, we prove several fine estimates involving convolutions of the so-called Clebsch-Gordan coefficients (which are elements of unitary matrices connecting reducible representations of SO(3)); this allows to intepret most of our asymptotic conditions in terms of coupling of angular momenta in a quantum mechanical system. Part of the proofs are based on recently established criteria for the weak convergence of multiple Wiener-It\^o integrals.