论文标题
随机超球谐波的非线性功能的总变化的收敛性
Convergence in Total Variation for nonlinear functionals of random hyperspherical harmonics
论文作者
论文摘要
随机的超球谐量是单位$ d $维球的高斯拉普拉斯本征函数（$ d \ ge 2 $）。我们研究其非线性统计量的总变异距离，即高能量极限，即用于拉普拉斯特征值的分化序列。 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 足够的。我们的论点需要对可能具有独立感兴趣的Gegenbauer多项式产物的矩量进行一些新的估计，我们通过图理论和图表公式之间的联系证明了这一点。
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-dimensional sphere ($d\ge 2$). 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.
