Association between Random Variables and the Dispersion of a Radon–Nikodym Derivative

Summary In a previous paper by one of the authors (Silvey, 1964) it was suggested that the Radon–Nikodym derivative of the joint distribution of two, not necessarily real-valued, random variables with respect to the product of their marginal distributions provides the analytic key for discussing association between the random variables. In this paper we shall develop this point and lend some support to the suggestion by proving that in the case of normal random vectors the distribution of this derivative, as determined by the product of the marginal distributions, becomes more widely spread out as association between the vectors increases. More precisely we shall show that the expected value of any continuous convex function of the derivative is a non-decreasing function of each of the canonical correlation coefficients between the two vectors.