  Maximum Mean Discrepancy (MMD) has been widely used in the areas of machine
learning and statistics to quantify the distance between two distributions in
the $p$-dimensional Euclidean space. The asymptotic property of the sample MMD
has been well studied when the dimension $p$ is fixed using the theory of
U-statistic. As motivated by the frequent use of MMD test for data of
moderate/high dimension, we propose to investigate the behavior of the sample
MMD in a high-dimensional environment and develop a new studentized test
statistic. Specifically, we obtain the central limit theorems for the
studentized sample MMD as both the dimension $p$ and sample sizes $n,m$ diverge
to infinity. Our results hold for a wide range of kernels, including popular
Gaussian and Laplacian kernels, and also cover energy distance as a special
case. We also derive the explicit rate of convergence under mild assumptions
and our results suggest that the accuracy of normal approximation can improve
with dimensionality. Additionally, we provide a general theory on the power
analysis under the alternative hypothesis and show that our proposed test can
detect difference between two distributions in the moderately high dimensional
regime. Numerical simulations demonstrate the effectiveness of our proposed
test statistic and normal approximation.
