2022/06/06 by Lucia Caramellino, Caramellino, Lucia, Giacomo Giorgio +3 · 2 citations
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Geometry and complex manifolds #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2206.02605
openalex publication_date 2022/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-dimensional sphere (d≥ 2). We study the convergence in Total Variation distance for their nonlinear statistics in the high energy limit, i.e., for diverging sequences of Laplace eigenvalues. Our approach takes advantage of a recent result by Bally, Caramellino and Poly (2020): combining the Central Limit Theorem in Wasserstein distance obtained by Marinucci and Rossi (2015) for Hermite-rank 2 functionals with new results on the asymptotic behavior of their Malliavin-Sobolev norms, we are able to establish second order Gaussian fluctuations in this stronger probability metric as soon as the functional is regular enough. Our argument requires some novel estimates on moments of products of Gegenbauer polynomials that may be of independent interest, which we prove via the link between graph theory and diagram formulas.