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