On a simple identity for the conditional expectation of orthogonal polynomials (Q1987713)

The author discusses the following theorem involving the Hermite polynomials [\textit{C. W. J. Granger} and \textit{P. Newbold}, J. R. Stat. Soc., Ser. B 38, 189--203 (1976; Zbl 0344.62076)]: let \(\left(X,Y \right)^T\) denote a two-dimensional random vector with a bivariate normal distribution with mean vector \(\left(0,0 \right)^T\) and covariance matrix \[ \Sigma= \left(\begin{array}{cc} 1 & \rho \\ \rho & 1\end{array}\right) \] and let \(H_{n}\) denote the \(n\)th Hermite polynomial, \(n=0,1,\ldots\). Then \[ E\left[ H_{n}(Y) | X \right]= \rho^{n} H_{n}(X)\, , \quad w. \, p.\, 1 \quad n=0,1,\ldots \] in order to explore a generalisation for further families of orthogonal polynomials. Namely, it is presented the following theorem. Let \(\left(X,Y \right)^T\) denote a two-dimensional random vector such that there exist orthogonal polynomials \(P_{0}\, , P_{1}\, , \ldots\) with respect to the marginal distribution of \(Y\) and there exist orthogonal polynomials \(Q_{0}\, , Q_{1}\, , \ldots\) with respect to the marginal distribution of \(X\). The following two conditions are equivalent. 1. For each \(r=0,1,\ldots\), \(E\left[ Y^r | X=x \right]\) and \(E\left[ X^r | Y=y \right]\) are polynomials of degree \(r\) in \(x\) and \(y\), respectively. 2. There exist nonzero constants \(a_{0},\, a_{1},\, \dots \) such that \[E\left[ P_{n}(Y) | X \right]= a_{n} Q_{n}(X)\, , \quad w. \, p.\, 1 \quad n=0,1,\ldots,\] and nonzero constants \(b_{0},\, b_{1},\, \dots \) such that \[E\left[ Q_{n}(X) | Y \right]= b_{n} P_{n}(Y)\, , \quad w. \, p.\, 1 \quad n=0,1,\ldots.\] An application to maximal correlation and a relationship to \(L^2\) completeness are also examined.