On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials

Thomas A. Severini*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 of a bivariate distribution are discussed.

Original languageEnglish (US)
Pages (from-to)13-27
Number of pages15
JournalSankhya A
Volume82
Issue number1
DOIs
StatePublished - Feb 1 2020

Keywords

  • Bivariate Dirichlet distribution
  • Bivariate gamma distribution
  • Jacobi polynomials
  • L completeness
  • Maximal correlation
  • Mehler’s identity
  • Primary 42C05
  • Secondary 60E05

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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