Stein-Malliavin approximations for nonlinear functionals of random eigenfunctions on \(\mathbb{S}^d\) (Q491505)

The main purpose of the present paper is to establish quantitative central limit theorems for the excursion volume of Gaussian eigenfunctions on \({\mathbb S}^d\), \(d \geq 2\). The authors provide a number of intermediate results of independent interest, namely the asymptotic analysis for the variance of moments of Gaussian eigenfunctions, the rates of convergence in various probability metrics for the so-called Hermite subordinated processes, and the analysis of arbitrary polynomials of finite order and square integrable nonlinear transforms. All these results could be useful to attack other problems, for instance quantitative central limit theorems for intrinsic volumes/Lipschitz-Killing curvatures of arbitrary order.