Two Necessary Conditions on the Representation of Bivariate Distributions by Polynomials

Abstract

Let $X$ and $Y$ be two unbounded random variables. Then two necessary conditions are proved concerning the structure of the bivariate distribution function of $X$ and $Y$ when it is expanded in the orthonormal polynomials of its marginal distributions. The first condition concerns the shrinking of the polynomial representation into a diagonal form, and the second is a generalization of the Sarmanov-Bratoeva theorem.