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We investigate Stein-Malliavin approximations for nonlinear functionals ofgeometric interest of Gaussian random eigenfunctions on the unit $d$-dimensional sphere ${\mathbb{S}}^{d},$ $d\geq 2.$ All our results areestablished in the high energy limit, i.e. for eigenfunctions corresponding togrowing eigenvalues. More precisely, we provide an asymptotic analysis for thevariance of random eigenfunctions, and also establish rates of convergence forvarious probability metrics for Hermite subordinated processes, arbitrarypolynomials of finite order and square integral nonlinear transforms; thelatter, for instance, allows to prove a quantitative Central Limit Theorem forthe excursion area. Some related issues were already considered in theliterature for the $2$-dimensional case ${\mathbb{S}}^{2}$; our results are newor improve the existing bounds even for this special case. Proofs are based onthe asymptotic analysis of moments of all order for Gegenbauer polynomials, andmake extensive use of the recent literature on so-called fourth-moment theoremsby Nourdin and Peccati.

Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on ${\mathbb{S}}^{d}$