Explicit expressions for joint moments of $n$-dimensional elliptical distributions

TL;DR

This paper derives new explicit formulas for joint moments of elliptical distributions, including normal, Student-t, logistic, and Laplace, using two different methods inspired by Stein's lemma, enhancing understanding of their expectations.

Contribution

It introduces two novel methods to compute joint moments of elliptical distributions, providing simplified formulas for various distributions and extending existing theoretical tools.

Findings

Derived new formulas for joint moments of elliptical distributions.

Provided simplified expressions for specific distributions like Student-t, logistic, and Laplace.

Enhanced theoretical understanding of expectations in elliptical distributions.

Abstract

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