Article

Title: On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials

Issue: Volume 82 Series A Part 1 Year 2020
Pages: 13 -- 27
Abstract
Consider a two-dimensional random vector (X, Y )T. Let Q0,Q1, . . . denote orthogonal polynomials with respect to the marginal distribution of X and let P0, P1, . . . denote orthogonal polynomials with respect to the marginal distribution of Y . In this paper, identities of the form E[Pn(Y ) |X] = anQn(X), for constants a0, a1, . . . are considered and necessary and sufficient conditions for this type of identity to hold are given,. The application of the identity to the maximal correlation of two random variables and to the L2 completeness ofa bivariate distribution are discussed.
Primary 42C05; Secondary 60E05
Keywords and phrases: Bivariate Dirichlet distribution, Bivariate gamma distribution, Jacobi polynomials, L2 completeness, Maximal correlation, Mehler’s identity 1