Group representations and high-resolution central limit theorems for subordinated spherical random fields

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 establish a new connection with random walks on the hypergroup $\widehat{\mathit{SO}(3)}$ (the dual of the group of rotations $SO(3)$), which mirrors analogous results previously established for fields defined on Abelian groups (see Marinucci and Peccati [Stochastic Process. Appl. 118 (2008) 585–613]). Our work is motivated by applications to cosmological data analysis.