  In this paper new tests for the independence of two high-dimensional vectors
are investigated. We consider the case where the dimension of the vectors
increases with the sample size and propose multivariate analysis of
variance-type statistics for the hypothesis of a block diagonal covariance
matrix. The asymptotic properties of the new test statistics are investigated
under the null hypothesis and the alternative hypothesis using random matrix
theory. For this purpose we study the weak convergence of linear spectral
statistics of central and (conditionally) non-central Fisher matrices. In
particular, a central limit theorem for linear spectral statistics of large
dimensional (conditionally) non-central Fisher matrices is derived which is
then used to analyse the power of the tests under the alternative.
  The theoretical results are illustrated by means of a simulation study where
we also compare the new tests with several alternative, in particular with the
commonly used corrected likelihood ratio test. It is demonstrated that the
latter test does not keep its nominal level, if the dimension of one sub-vector
is relatively small compared to the dimension of the other sub-vector. On the
other hand the tests proposed in this paper provide a reasonable approximation
of the nominal level in such situations. Moreover, we observe that one of the
proposed tests is most powerful under a variety of correlation scenarios.
