Group representations and high frequency asymptotics for subordinated
random fields on the sphere.

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 hypergroup $\widehat{SO\left( 3\right) }$ (the dual of
SO(3)), which mirrors analogous results previously established for
fields defined on Abelian groups. Our work is motivated by
applications to cosmological data analysis, and specifically by the
probabilistic modelization 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-Ito
integrals.


The paper is based on joint work with G.Peccati (University Paris
VI).