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On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials

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.

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Acknowledgements

I would like to thank the associate editor and referees for a number of useful comments.

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Correspondence to Thomas A. Severini.

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Severini, T.A. On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials. Sankhya A 82, 13–27 (2020). https://doi.org/10.1007/s13171-018-00161-0

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  • DOI: https://doi.org/10.1007/s13171-018-00161-0

Keywords

  • Bivariate Dirichlet distribution
  • Bivariate gamma distribution
  • Jacobi polynomials
  • L2 completeness
  • Maximal correlation
  • Mehler’s identity

AMS (2000) subject classification.

  • Primary 42C05
  • Secondary 60E05