One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix

$(\frac{1}{\sqrt{n}} X_n - zI)(\frac{1}{\sqrt{n}} X_n - zI)^\ast$,

where $X_n$ is an $n \times n$ matrix with i.i.d. square integrable coefficients.

This is done with the Stieltjes transform, however Bai and Silverstein remark in their book that (under some assumptions on the random variables), one can also prove that moments of the empirical distribution converge almost surely to $\mu_k(|z|^2)$ (they leave the details as an exercise).

It is indeed not difficult to show that the moments converge to a number which depends only on $|z|^2$. My question is whether there is a nice formula for $\mu_k(|z|^2)$. For z = 0, we have just the Marchenko-Pastur distribution and there is a relatively good-looking formula involving binomial coefficients. For other values of $z$ we have some additional terms corresponding to some graphs (depending on $k$). This is enough to provide a moment proof of the convergence of the empirical spectral distribution, but I wonder if there is a nicer formula for the moments, e.g. involving only binomial coefficients and algebraic operations (no summation over combinatorial objects).


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