发表机构
Karlsruhe Institute of Technology; German Cancer Research Center; Université du Québec à Trois-Rivières; Pennsylvania State University（卡尔斯鲁厄理工学院; 德国癌症研究中心; 魁北克大学三河分校; 宾夕法尼亚州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整求解了BHEP多元正态性检验中Henze-Zirkler与Henze-Wagner算子在所有维度和平滑参数下的谱，证明了二者非零特征值一致，并刻画了零空间与特征值方程。
AI 中文摘要
Baringhaus-Henze-Epps-Pulley（BHEP）多元正态性检验是基于经验特征函数与高斯特征函数之间的高斯加权$L^2$距离的仿射不变拟合优度检验。1990年，Henze和Zirkler通过标准高斯空间上积分算子的特征值表达了极限零分布。1997年，Henze和Wagner获得了更简单的协方差核，并提出了计算高斯加权空间上所得算子特征值的问题。尽管后续工作处理了单变量情形和少数低维数值近似，但完整的全维谱问题仍未解决。本文确定了每个维度$d \in \mathbb{N}$和每个平滑参数$\beta > 0$的两个完整谱。这两个算子被证明具有$\mathcal{X}_{\beta,d}^*\mathcal{X}_{\beta,d}$和$\mathcal{X}_{\beta,d}\mathcal{X}_{\beta,d}^*$的形式，其中$\mathcal{X}_{\beta,d}$是同一个Hilbert-Schmidt算子。因此，它们的非零特征值一致（包括重数），而Henze-Zirkler算子的零空间被精确识别。Henze-Wagner分解中的高斯积分算子通过Mehler公式对角化，旋转对称性将有限秩修正限制在与球谐函数度数$0$、$1$和$2$相关的扇区。度数$1$和度数$2$的特征值由标量超越方程刻画，径向特征值由显式的极点安全Fredholm行列式刻画。本文确立了非负性、重数、特征函数重构、完备性、迹恒等式以及所有例外极点情形的完整刻画。
英文摘要
The Baringhaus-Henze-Epps-Pulley (BHEP) tests for multivariate normality are affine-invariant goodness-of-fit tests based on a Gaussian-weighted $L^2$ distance between empirical and Gaussian characteristic functions. In 1990, Henze and Zirkler expressed the limiting null distribution through the eigenvalues of an integral operator on the standard Gaussian space. In 1997, Henze and Wagner obtained a simpler covariance kernel and raised the problem of calculating the eigenvalues of the resulting operator on a Gaussian-weighted space. Although subsequent work treated the univariate case and numerical approximations in a few low dimensions, the complete all-dimensional spectral problem remained open. This paper determines both complete spectra for every dimension $d \in \mathbb{N}$ and every smoothing parameter $β> 0$. The two operators are shown to have the forms $\mathcal{X}_{β,d}^*\mathcal{X}_{β,d}$ and $\mathcal{X}_{β,d}\mathcal{X}_{β,d}^*$ for the same Hilbert-Schmidt operator $\mathcal{X}_{β,d}$. Consequently, their nonzero eigenvalues agree, including multiplicities, while the null space of the Henze-Zirkler operator is identified exactly. The Gaussian integral operator in the Henze-Wagner decomposition is diagonalized by Mehler's formula, and rotational symmetry confines the finite-rank correction to the sectors associated with spherical harmonics of degrees $0$, $1$, and $2$. The degree-$1$ and degree-$2$ eigenvalues are characterized by scalar transcendental equations, and the radial eigenvalues by an explicit pole-safe Fredholm determinant. The paper establishes nonnegativity, multiplicities, eigenfunction reconstruction, completeness, the trace identity, and a complete characterization of all exceptional pole cases.
Comments52 pages, 3 figures, 5 tables
