On the Limit Distribution of the Canonical Correlation Coefficients Between the Past and the Future of a High-Dimensional White Noise

It is shown that the distribution of the estimated canonical correlation coefficients between the past and the future of a high-dimensional multivariate white noise sequence converges almost surely towards a limit distribution whose density is given in closed form. A sketch of proof, based on free probability technics, is provided. Finally, it is briefly explained how this result can be used to produce consistent uncorrelatedness tests in the high-dimensional context.