AI 中文总结
本文针对非退化U统计量的希尔伯特空间范数，建立了非渐近Berry–Esseen界与三类自助近似，得到的检验具渐近正确性且分离率达极小极大最优。
AI 中文摘要
我们针对非退化U统计量的希尔伯特空间范数建立了非渐近的Berry–Esseen界与自助近似。本文的三角阵列框架允许核函数取值于可分希尔伯特空间$H_n$，该空间可随样本量变化，因此涵盖了无限维空间及维数递增的欧氏空间。高斯近似结合了希尔伯特空间版本的Stein方法与Hoeffding分解中高阶项的精细控制，其误差取决于核函数的四阶矩及Hájek投影协方差算子的谱几何，尤其是前主特征方向外的平方谱质量占比。因此，只要前主方向外仍有足够谱质量，该理论可适配奇异及近似低秩的协方差算子。当$H_n=\mathbb{R}^{d_n}$时，我们在逐坐标四阶矩条件下得到了显式依赖维数的界；还针对经验、高斯加权及刀切乘子自助法建立了近似界，得到了具有渐近正确尺寸的检验，且对协方差依赖分离率下的备择假设具有一致性。针对成对Kendall's tau系数向量的检验，匹配的极小极大下界表明，所得分离率在稠密高斯相关备择假设下达到极小极大速率最优。
英文摘要
We establish non-asymptotic Berry--Esseen bounds and bootstrap approximations for the Hilbert-space norm of nondegenerate $U$-statistics. Our triangular-array framework allows the kernel to take values in a separable Hilbert space $H_n$ that may vary with the sample size, thereby covering both infinite-dimensional spaces and Euclidean spaces of increasing dimension. The Gaussian approximation combines a Hilbert-space version of Stein's method with refined control of the higher-order terms in the Hoeffding decomposition. Its error depends on fourth moments of the kernel and on the spectral geometry of the covariance operator of the Hájek projection, particularly the proportion of squared spectral mass lying outside the leading eigendirection. Consequently, the theory accommodates singular and approximately low-rank covariance operators, provided that sufficient spectral mass remains beyond the leading direction. For $H_n=\mathbb{R}^{d_n}$, we obtain dimension-explicit bounds under coordinatewise fourth-moment conditions. We also establish approximation bounds for the empirical, Gaussian-weighted, and jackknife multiplier bootstraps, yielding tests with asymptotically correct size and consistency against alternatives at covariance-dependent separation rates. For testing the vector of pairwise Kendall's tau coefficients, a matching minimax lower bound shows that the resulting separation rate is minimax rate-optimal over dense Gaussian correlation alternatives.
