# Trace of product of two Wishart matrices

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 $$X$$?

• 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. Commented Sep 6, 2023 at 12:45
• @YemonChoi I've made some edits. Commented Sep 6, 2023 at 19:42

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 $${\rm Var}\,[X]=r^2+2r+{\cal O}(n^{-1}).$$

• 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? Commented Sep 7, 2023 at 0:13
• higher cumulants will be smaller by powers of $1/n$. Commented Sep 7, 2023 at 5:35