Gaussian Approximation for High-Dimensional Weighted and Two-Sample $U$-statistics with Size-Dependent Kernels

Abstract

We develop Gaussian approximations for high-dimensional vectors formed by second-order $U$-and $V$-statistics whose kernels depend on sample size. To accommodate weighted and two-sample $U$-and $V$-statistics within a unified framework, we establish our results under independent but not identically distributed (i.n.i.d.) sampling. The resulting theorems apply to a broad range of problems, including estimation in linear regression models with many covariates or many-weak instruments, variance component analysis with growing ranks, smoothed two-sample testing, and inference for separately exchangeable arrays, among others. Our results do not require either component of the Hoeffding decomposition to dominate, thereby covering both non-degenerate and degenerate regimes. In addition, we develop maximal inequalities for high-dimensional $U$-statistics with size-dependent kernels under the i.n.i.d.~setting, in a form that remains sharp across a broad range of applications, which may be of independent interest. Finally, applications to estimation and testing with many weak instrumental variables demonstrate that, our results are sharp enough to recover the corresponding fixed-dimensional rate conditions up to logarithmic factors.