Generalized form of Price´s theorem and its converse

The case of unity-variance random variables governed by the joint probability density is considered, where the density depends on the (normalized) cross-covariances . It is shown that the condition holds for an "arbitrary" function of variables if and only if the underlying density is the usual -dimensional Gaussian density for correlated random variables. This result establishes a generalized form of Price\´s theorem in which: 1) the relevant condition subsumes Price\´s original condition; 2) the proof is accomplished without appeal to Laplace integral expansions; and 3) conditions referring to derivatives with respect to diagonal terms are avoided, so that the unity variance assumption can be retained.