Expressions for joint moments of elliptical distributions

Inspired by Stein’s lemma, we derive two expressions for the joint moments of elliptical distributions. We use two different methods to derive E [ X 1 2 f ( X ) ] for any measurable function f satisfying some regularity conditions. Then, by making use of this result, we obtain new formulae for expectations of product of normally distributed random variables, and also present simplified expressions of E [ X 1 2 f ( X ) ] for multivariate Student- t , logistic and Laplace distributions.