On bootstrap approximations for high-dimensional U-statistics and random quadratic forms

This paper establishes a unified theory of bootstrap approximations for the probabilities of a non-degenerate U-statistic belonging to the hyperrectangles in $\mathbb{R}^d$ when $d$ is large. Specifically, we show that the empirical bootstrap with the multinomial weights and the randomly reweighted bootstrap with iid Gaussian weights can be viewed as a more general random quadratic form and they are inferentially first-order equivalent in the following sense. Subject to mild moment conditions on the kernel, both methods achieve the same uniform rate of convergence over all $d$-dimensional hyperrectangles. In particular, they are asymptotically valid with high probability when the dimension $d$ can be as large as $O(e^{n^c})$ for some constant $c \in (0,1/7)$. We also establish their equivalence to a Gaussian multiplier bootstrap with the jackknife covariance matrix estimator of the U-statistics. The bootstrap limit theorems rely on a general Gaussian approximation result and the tail probability inequalities for the maxima of non-degenerate U-statistics with the unbounded kernel. In particular, we derive explicit non-asymptotic error bounds for uniform approximation over the class of hyperrectangles in $\mathbb{R}^d$. Results established in this paper are nonlinear generalizations of the Gaussian and bootstrap approximations of the maxima of the high-dimensional sample mean vectors to the U-statistics of order two.