A note on Stein's lemma for multivariate elliptical distributions

• We provide a simple proof for Stein's lemma.
• The proof is based on the idea of conditioning and the use of linear regression.
• We also provide an instructive proof for the classic linear regression result.

When two random variables are bivariate normally distributed Stein's original lemma allows to conveniently express the covariance of the first variable with a function of the second. Landsman and Neslehova (2008) extend this seminal result to the family of multivariate elliptical distributions. In this paper we use the technique of conditioning to provide a more elegant proof for their result. In doing so, we also present a new proof for the classical linear regression result that holds for the elliptical family.