An Isserlis theorem for Elliptical Distributions?

Question

For my purposes, I am only interested in rotationally invariant Elliptical Distributions. So suppose $x$ is a $p$-dimensional random variable with zero mean and identity covariance, with density equal to $g(x^{\top}x)$. I am interested in third and fourth moments: $$ E\left[x_i x_j x_k \right] \quad\mbox{and}\quad E\left[x_i x_j x_k x_l \right], $$ where the indices may be duplicated. I have some hints from this paper by Kan, but not enough.

Answer 1

Following C. Vignat, S. Bhatnagar, as suggested by BGM, if we let $X = a \Sigma^{1/2} \frac{Z}{\left\|Z\right\|}$, where $Z$ is an $n$-dimensional multivariate normal with zero mean and identity covariance, and where $a$ is some random variable that effectively controls the norm of $X$, then $$ E\left[x_i x_j\right] = \frac{E\left[a^2\right]}{n} \sigma_{i,j}, $$ and thus the covariance of $X$ is $$ \frac{E\left[a^2\right]}{n}\Sigma. $$ Moving on to higher order moments we have $$ E\left[x_i x_j x_k\right] = 0, $$ and $$ E\left[x_i x_j x_k x_l\right] = \frac{E\left[a^4\right]}{n\left(n+2\right)}\left(\sigma_{i,j}\sigma_{k,l} + \sigma_{i,k}\sigma_{j,l} + \sigma_{i,l}\sigma_{j,k} \right). $$

For raw, or uncentered moments, consider let $X = \mu + a \Sigma^{1/2} \frac{Z}{\left\|Z\right\|}$. Then $$ E\left[x_i x_j\right] = \mu_i\mu_j + \frac{E\left[a^2\right]}{n} \sigma_{i,j}, $$ $$ E\left[x_i x_j x_k\right] = \mu_i\mu_j\mu_k + \frac{E\left[a^2\right]}{n} \left(\mu_i \sigma_{j,k} + \mu_j \sigma_{i,k} + \mu_k \sigma_{i,j}\right), $$ and \begin{array}\, E \left[x_i x_j x_k x_l\right] &= \mu_i\mu_j\mu_k\mu_l \\ &\phantom{=}+\frac{E\left[a^2\right]}{n} \left(\mu_i \mu_j\sigma_{k,l} + \mu_i\mu_k \sigma_{j,l} + \mu_i\mu_l \sigma_{j,k} + \mu_j \mu_k\sigma_{i,l} + \mu_j\mu_l \sigma_{i,k} + \mu_k\mu_l \sigma_{i,j} \right) \\ &\phantom{=}+\frac{E\left[a^4\right]}{n\left(n+2\right)}\left(\sigma_{i,j}\sigma_{k,l} + \sigma_{i,k}\sigma_{j,l} + \sigma_{i,l}\sigma_{j,k} \right). \end{array}