Asymptotic Behaviour of Level Sets of Needlet Random Fields

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.

Flexible-bandwidth Needlets

“…Needlets on one hand represent a tightframe system and hence satisfy classical requirements of approximation theory; on the other hand under some regularity conditions needlet coefficients have been shown to enjoy asymptotic uncorrelation properties (in the high-resolution sense) which makes their application to random fields extremely powerful. Extensions of the needlet construction to more general homogeneous spaces of compact groups were given for instance by [10,16,20]; statistical applications are currently too many to be recalled in any reasonable completeness: we refer for instance to [18,19] or more recently [6,9,11,12,22,36,38,14,23]. Applications in Cosmology and Astrophysics are discussed for instance in [7,27,32,35,39,40]) and the references therein.…”