Mixing coefficient, generalized maximal correlation coefficients, and weakly positive measures

Let [mu] be a positive finite Borel measure on the real line R. For t >= 0 let et · E1 and E2 denote, respectively, the linear spans in L2(R, [mu]) of {eisx, s > t} and {eisx, s C such that theta; = 1, denote by [alpha]t([theta], [mu]) the angle between [theta] · et · E1 and E2. The problems considered here are that of describing those measures [mu] for which (1) [alpha]t([theta], [mu]) > 0, (2) [alpha]t([theta], [mu]) --> [pi]/2 as t --> [infinity] (such [mu] arise as the spectral measures of strongly mixing stationary Gaussian processes), and (3) give necessary and sufficient conditions for the rate of convergence of the generalized maximal correlation coefficient: [varrho]t([theta], [mu]) = cos [alpha]t([theta], [mu]). Using this coefficient we characterize the stationary continuous processes that are (a) completely regular and (b) strongly mixing Gaussian. We also give necessary and sufficient conditions for the rate of convergence of (a) the maximal correlation coefficient and (b) the mixing coefficient in the Gaussian case.