COVARIANCE INEQUALITIES FOR STRONGLY MIXING PROCESSES

Let X and Y be two real-valued random variables. Let alpha denote the strong mixing coefficient between the two sigma-fields generated respectively by X and Y, and Q(X) (u) = inf {t: P (Absolute value of X > t) less-than-or-equal-to u} be the quantile function of Absolute value of X. We prove the following new covariance inequality: \Cov (X, Y)\ less-than-or-equal-to 2 integral-2alpha/0 Q(X) (u) Q(Y) (u) du, which we show to be sharp, up to a constant factor. We apply this inequality to improve on the classical bounds for the variance of partial sums of strongly mixing processes.