Sup-norm asymptotics of high-dimensional matrix-variate U-statistics and its applications

We study the asymptotics of high-dimensional U-statistics of order two under the supremum norm. Sharp expectation bound and higher-order moments inequalities of matrix-variate U-statistics with unbounded kernel are established. The estimates involve "mixed norms" of the matrix kernels expressed in terms of maxima of empirical processes. For non-degenerate U-statistics, we propose a two-step Gaussian approximation procedure and derive its convergence rate that decays polynomially in sample size. We also supplement a practical Gaussian wild bootstrap method to approximate the quantiles of the maxima of centered U-statistics. Our theoretical results are demonstrated on several statistical applications involving adaptive tuning parameter selection, simultaneous inference and related functional estimation of the covariance matrix for high-dimensional non-Gaussian data.