Correlations of functions of normal variables

Let (X1, X2,..., Xk, Y1, Y2,..., Yk) be multivariate normal and define a matrix C by Cij = cov(Xi, Yj). If (i) (X1,..., Xk) = (Y1,..., Yk) and (ii) C is symmetric positive definite, then 0 < varf(X1,..., Xk) < [infinity] => corr(f(X1,..., Xk),f(Y1,..., Yk)) > 0. Condition (i) is necessary for the conclusion. The sufficiency of (i) and (ii) follows from an infinite-dimensional version, which can also be applied to a pair of jointly normal Brownian motions.