Let $A,B$ be two independent complex Wishart matrices, $A,B\sim CW_p(\mathbf{I},n)$, that is $A=\frac1n GG^\dagger$& $B=\frac1n QQ^\dagger$ where $G$ and $Q$ are independent $p\times n$ complex Gaussian matrices with i.i.d. Gaussian entries of zero mean and unit variance. Let $X:=\operatorname{Tr} AB$. What's the probability distribution of $X$?
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$\begingroup$ Assuming that your notation is the same as en.wikipedia.org/wiki/Wishart_distribution - can't you reformulate your question purely in terms of sums of products of IID complex Gaussians? This might help readers to give an answer. $\endgroup$– Yemon ChoiCommented Sep 6, 2023 at 12:45
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$\begingroup$ @YemonChoi I've made some edits. $\endgroup$– ShadumuCommented Sep 6, 2023 at 19:42
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In the limit $n,p\rightarrow\infty$, at fixed ratio $r=p/n$, the probability distribution of $$X=n^{-2}\,{\rm Tr}\,GG^\dagger QQ^\dagger=n^{-2}\sum_{i,k=1}^p\sum_{j,l=1}^n G_{ij}\bar{G}_{kj}Q_{kl}\bar{Q}_{il},$$ tends to a Gaussian, so it would be sufficient to find the first two moments. The mean is given by $$\mathbb{E}[X]=n^{-2}\sum_{i=1}^p\sum_{j,l=1}^n\mathbb{E}[|G_{ij}|^2]\mathbb{E}[|Q_{il}|^2]=nr.$$ The calculation of the second moment is a bit more lengthy, I find the variance $${\rm Var}\,[X]=r^2+2r+{\cal O}(n^{-1}).$$
answered Sep 6, 2023 at 21:11
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$\begingroup$ Thanks! How quickly does X tend to a Gaussian? If one examines the higher moments, say nth or pth moments, could one tell its difference from Gaussian? $\endgroup$– ShadumuCommented Sep 7, 2023 at 0:13
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$\begingroup$ higher cumulants will be smaller by powers of $1/n$. $\endgroup$ Commented Sep 7, 2023 at 5:35