A matricial extension of the Helson-Sarason theorem and a characterization of some multivariate linearly completely regular processes

We generalize the theorems of Helson-Szegö and Helson-Sarason for matricial measures. We study two-weighted inequalities for the Hilbert transform in 0, 2π] and in R and give a characterization for the positivity of the angle between past and future of multivariate weakly stationary stochastic processes, in the discrete and the continuous case. We also characterize the multivariate weakly stationary stochastic processes that are linearly completely regular and study the rate of convergence of the maximal correlation coefficient.