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This paper proposes a new statistic to test independence between two highdimensional random vectors ${\mathbf{X}}:p_1\times1$ and${\mathbf{Y}}:p_2\times1$. The proposed statistic is based on the sum ofregularized sample canonical correlation coefficients of ${\mathbf{X}}$ and${\mathbf{Y}}$. The asymptotic distribution of the statistic under the nullhypothesis is established as a corollary of general central limit theorems(CLT) for the linear statistics of classical and regularized sample canonicalcorrelation coefficients when $p_1$ and $p_2$ are both comparable to the samplesize $n$. As applications of the developed independence test, various types ofdependent structures, such as factor models, ARCH models and a generaluncorrelated but dependent case, etc., are investigated by simulations. As anempirical application, cross-sectional dependence of daily stock returns ofcompanies between different sections in the New York Stock Exchange (NYSE) isdetected by the proposed test.

Independence test for high dimensional data based on regularized canonical correlation coefficients