  We give two asymptotic results for the empirical distance covariance on
separable metric spaces without any iid assumption on the samples. In
particular, we show the almost sure convergence of the empirical distance
covariance for any measure with finite first moments, provided that the samples
form a strictly stationary and ergodic process. We further give a result
concerning the asymptotic distribution of the empirical distance covariance
under the assumption of absolute regularity of the samples and extend these
results to certain types of pseudometric spaces. In the process, we derive a
general theorem concerning the asymptotic distribution of degenerate
V-statistics of order 2 under a strong mixing condition.
