  This paper proposes a new mutual independence test for a large number of high
dimensional random vectors. The test statistic is based on the characteristic
function of the empirical spectral distribution of the sample covariance
matrix. The asymptotic distributions of the test statistic under the null and
local alternative hypotheses are established as dimensionality and the sample
size of the data are comparable. We apply this test to examine multiple MA(1)
and AR(1) models, panel data models with some spatial cross-sectional
structures. In addition, in a flexible applied fashion, the proposed test can
capture some dependent but uncorrelated structures, for example, nonlinear
MA(1) models, multiple ARCH(1) models and vandermonde matrices.
  Simulation results are provided for detecting these dependent structures. An
empirical study of dependence between closed stock prices of several companies
from New York Stock Exchange (NYSE) demonstrates that the feature of
cross--sectional dependence is popular in stock markets.
