  We apply the concept of distance covariance for testing independence of two
long-range dependent time series. As test statistic we propose a linear
combination of empirical distance cross-covariances. We derive the asymptotic
distribution of the test statistic, and we show consistency against arbitrary
alternatives. The asymptotic theory developed in this paper is based on a novel
non-central limit theorem for stochastic processes with values in an
$L^2$-Hilbert space. This limit theorem is of general theoretical interest
which goes beyond the context of this article. Subject to the dependence in the
data, the standardization and the limit distributions of the proposed test
statistic vary. Since the limit distributions are unknown, we propose a
subsampling procedure to determine the critical values for the proposed test,
and we provide a proof for the validity of subsampling. In a simulation study,
we investigate the finite-sample behavior of our test, and we compare its
performance to tests based on the empirical cross-covariances. As an
application of our results we analyze the cross-dependencies between mean
monthly discharges of three rivers.
