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