Distance covariance is a popular dependence measure for two random vectors $X$ and $Y$ of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that $X$ and $Y$ are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert-Schmidt) kernel distance covariance in high dimensions, primarily in the Gaussian case. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between $X$ and $Y$. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by $n\cdot \text{dcor}^2(X,Y)/\sqrt{2}$ in the high dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between $X$ and $Y$.
翻译:远程常量是两种随机矢量的流行依赖度度度, 可能具有不同维度和类型。 近些年来, 文献中集中努力在高维环境下理解样本中距离常量的分布特性, 专门强调美元和美元之间的独立是独立的无效情况。 本文给出了样本距离常量的第一个非核中心限值, 以及更普通的样本( Hilbert- Schmidt) 在高维度( Gausian ) 的情况下, 内端距离比值可能不同维度和类型。 新的非核中央中位限制使得在高维度环境中, 样本中位距离差值的分布性能公式在高维度环境中, 广泛使用的通用内核距离相关测试在$X美元和Y美元之间。 权力公式特别揭示了一个有趣的普遍性现象: 普通内层距离相关测试的力量完全由 $\ cdott\ text {dcrickral 2( Hyral) ral- dreal deal deal lader)。 在高空端的内端值中位值中, 和内端值的内端值的内端值和内端值的内端值之间, 显示的内端值的内端值和内端值的内端值的内端值的内端距值和内端距值的内端距值的值的内端值之间, 。