Kolmogorov'e remark on the Hotelling canonical correlations

An alternative method of proof is given for the derivation of the Hotelling canonical correlations in the multivariate normal distribution. The same method is used to prove that in the bivariate case, the marginal variables have greater correlation than any other pair of functions of the form, g(X) and h(Y), in the merginal veriables, X and Y. This is a specia1 case of a proposition due to Kolmogorov that the Hotelling solution for functions linear in the marginal variables is also a solution for all pairs of functions of the sets of variables, which are of finite variance and of the form, g(X1, X2, …, Xp) and h(Y1, Y2, …, Yq), the variables X1, X2, …, Xp, Y1, Y2, …, Yq being jointly normal.