Let $X$ and $Y$ be independent random symmetric matrices. What can one say about $\mathbb{E} [X Y X Y]$ or $\mathrm{trace} \mathbb{E} [X Y X Y]$ in terms of properties of $X$ and $Y$?


In particular, can we compute such expectations if $X$ is Wishart distributed and $Y$ is inverse-Wishart distributed (i.e. the inverse of $Y$ is Wishart distributed)?



Expectation of product of random matrices
You could compute these expectation values from the known marginal distribution of the matrix elements of the Wishart and inverse Wishart ensembles; as a simpler test case here is the answer for a single product $XY$ of a $(p,n)$ Wishart and a $(p,\nu)$ inverse Wishart matrix (with identity scale matrices):
$${\rm tr}\,\mathbb{E}[XY]=\sum\_{k\_1,k\_2=1}^{p}\mathbb{E}[X\_{k\_1k\_2}]\mathbb{E}[Y\_{k\_2k\_1}]=\frac{pn}{\nu-p-1}.$$


