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We prove a convergence theorem for U-statistics of degree two, where the datadimension $d$ is allowed to scale with sample size $n$. We find that thelimiting distribution of a U-statistic undergoes a phase transition from thenon-degenerate Gaussian limit to the degenerate limit, regardless of itsdegeneracy and depending only on a moment ratio. A surprising consequence isthat a non-degenerate U-statistic in high dimensions can have a non-Gaussianlimit with a larger variance and asymmetric distribution. Our bounds are validfor any finite $n$ and $d$, independent of individual eigenvalues of theunderlying function, and dimension-independent under a mild assumption. As anapplication, we apply our theory to two popular kernel-based distributiontests, MMD and KSD, whose high-dimensional performance has been challenging tostudy. In a simple empirical setting, our results correctly predict how thetest power at a fixed threshold scales with $d$ and the bandwidth.

A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing