  We present a test for independence of two strictly stationary time series
based on a bootstrap procedure for the distance covariance. Our test detects
any kind of dependence between the two time series within an arbitrary maximum
lag $L$. In simulation studies, our test outperforms alternative testing
procedures. In proving the validity of the underlying bootstrap procedure, we
generalise bounds for the Wasserstein distance between an empirical measure and
its marginal distribution under the assumption of $\alpha$-mixing. Previous
results of this kind only existed for i.i.d. processes.
