Remarks on Uncorrelated Gaussian Dependent Random Variables

Formula (3) of the present paper gives a general example of two uncorrelated Gaussian dependent random variables with non-Gaussian joint distribution. In this formula $\varphi(x)$ is a bounded even real valued function defined on the real line such that $\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty\varphi(x)e^{-x^2/2}\,dx=0$, $h$ is equal to $\sup_{-\infty<x<\infty}|\varphi(x)|$ and $-1\le\lambda\le1$.