  In this paper, we are concerned with the independence test for $k$
high-dimensional sub-vectors of a normal vector, with fixed positive integer
$k$. A natural high-dimensional extension of the classical sample correlation
matrix, namely block correlation matrix, is raised for this purpose. We then
construct the so-called Schott type statistic as our test statistic, which
turns out to be a particular linear spectral statistic of the block correlation
matrix. Interestingly, the limiting behavior of the Schott type statistic can
be figured out with the aid of the Free Probability Theory and the Random
Matrix Theory. Specifically, we will bring the so-called real second order
freeness for Haar distributed orthogonal matrices, derived in \cite{MP2013},
into the framework of this high-dimensional testing problem. Our test does not
require the sample size to be larger than the total or any partial sum of the
dimensions of the $k$ sub-vectors. Simulated results show the effect of the
Schott type statistic, in contrast to those of the statistics proposed in
\cite{JY2013} and \cite{JBZ2013}, is satisfactory. Real data analysis is also
used to illustrate our method.
