Spectral Analysis of Gaussian Integral Operators Arising in BHEP Tests
Abstract
The Baringhaus-Henze-Epps-Pulley (BHEP) tests for multivariate normality are affine-invariant goodness-of-fit tests based on a Gaussian-weighted L2 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 ∈ ℕ and every smoothing parameter β> 0. The two operators are shown to have the forms 𝒳β,d*𝒳β,d and 𝒳β,d𝒳β,d* for the same Hilbert-Schmidt operator 𝒳β,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.
Related
- Two pointwise estimates for singular integral operators over p-adic groups
- Weak type (1,1) boundedness of noncommutative singular integral operators with rough kernels
- Essentially semi-commuting singular integral operators with Cauchy kernel on L2
- An integral representation of eigenfunctions for the deformed Noumi--Sano operators
- Multilinear rough singular integral operators may not be of the weak type at the endpoint
BibTeX
@misc{ebner2026spectral,
title = {{Spectral Analysis of Gaussian Integral Operators Arising in BHEP Tests}},
author = {Bruno Ebner and Dominic Edelmann and Norbert Henze and Frédéric Ouimet and Donald Richards},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.28464},
eprint = {2609.28464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28464},
}
RIS
TY - UNPB
TI - Spectral Analysis of Gaussian Integral Operators Arising in BHEP Tests
AU - Bruno Ebner
AU - Dominic Edelmann
AU - Norbert Henze
AU - Frédéric Ouimet
AU - Donald Richards
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.28464
UR - https://arxiv.org/abs/2609.28464
AB - 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 ∈ ℕ and every smoothing parameter β> 0. The two operators are shown to have the forms 𝒳_β,d^*𝒳_β,d and 𝒳_β,d𝒳_β,d^* for the same Hilbert-Schmidt operator 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.
ER -