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This paper studies the Gaussian approximation of high-dimensional andnon-degenerate U-statistics of order two under the supremum norm. We propose atwo-step Gaussian approximation procedure that does not impose structuralassumptions on the data distribution. Specifically, subject to mild momentconditions on the kernel, we establish the explicit rate of convergence thatdecays polynomially in sample size for a high-dimensional scaling limit, wherethe dimension can be much larger than the sample size. We also supplement apractical Gaussian wild bootstrap method to approximate the quantiles of themaxima of centered U-statistics and prove its asymptotic validity. The wildbootstrap is demonstrated on statistical applications for high-dimensionalnon-Gaussian data including: (i) principled and data-dependent tuning parameterselection for regularized estimation of the covariance matrix and its relatedfunctionals; (ii) simultaneous inference for the covariance and rankcorrelation matrices. In particular, for the thresholded covariance matrixestimator with the bootstrap selected tuning parameter, we show that theGaussian-like convergence rates can be achieved for heavy-tailed data, whichare less conservative than those obtained by the Bonferroni technique thatignores the dependency in the underlying data distribution. In addition, wealso show that even for subgaussian distributions, error bounds of thebootstrapped thresholded covariance matrix estimator can be much tighter thanthose of the minimax estimator with a universal threshold.

Gaussian approximation for the sup-norm of high-dimensional matrix-variate U-statistics and its applications