  In this paper, we introduce a ${\mathcal L}_2$ type test for testing mutual
independence and banded dependence structure for high dimensional data. The
test is constructed based on the pairwise distance covariance and it accounts
for the non-linear and non-monotone dependences among the data, which cannot be
fully captured by the existing tests based on either Pearson correlation or
rank correlation. Our test can be conveniently implemented in practice as the
limiting null distribution of the test statistic is shown to be standard
normal. It exhibits excellent finite sample performance in our simulation
studies even when the sample size is small albeit dimension is high, and is
shown to successfully identify nonlinear dependence in empirical data analysis.
On the theory side, asymptotic normality of our test statistic is shown under
quite mild moment assumptions and with little restriction on the growth rate of
the dimension as a function of sample size. As a demonstration of good power
properties for our distance covariance based test, we further show that an
infeasible version of our test statistic has the rate optimality in the class
of Gaussian distribution with equal correlation.
