  Distance covariance is a popular dependence measure for two random vectors
$X$ and $Y$ of possibly different dimensions and types. Recent years have
witnessed concentrated efforts in the literature to understand the
distributional properties of the sample distance covariance in a
high-dimensional setting, with an exclusive emphasis on the null case that $X$
and $Y$ are independent. This paper derives the first non-null central limit
theorem for the sample distance covariance, and the more general sample
(Hilbert-Schmidt) kernel distance covariance in high dimensions, primarily in
the Gaussian case. The new non-null central limit theorem yields an
asymptotically exact first-order power formula for the widely used generalized
kernel distance correlation test of independence between $X$ and $Y$. The power
formula in particular unveils an interesting universality phenomenon: the power
of the generalized kernel distance correlation test is completely determined by
$n\cdot \text{dcor}^2(X,Y)/\sqrt{2}$ in the high dimensional limit, regardless
of a wide range of choices of the kernels and bandwidth parameters.
Furthermore, this separation rate is also shown to be optimal in a minimax
sense. The key step in the proof of the non-null central limit theorem is a
precise expansion of the mean and variance of the sample distance covariance in
high dimensions, which shows, among other things, that the non-null Gaussian
approximation of the sample distance covariance involves a rather subtle
interplay between the dimension-to-sample ratio and the dependence between $X$
and $Y$.
