Non-asymptotic results for bulk of random Wishart matrix

Question

Let $n$ be a positive integer and let $X_n$ be an $n\times n$ random matrix whose entries are iid standard gaussian random variables. I am interested in the distribution of the average singular value of $\frac1{\sqrt{n}}X_n$.

Let $\alpha_n$ be the expected value of the average of the singular values of $\frac1{\sqrt{n}}X_n$. $\alpha_1$ is easy to compute and is equal to $\sqrt{\frac2\pi}$. It is also easy to understand the $n\to\infty$ limit of this distribution because $W_n = \frac1n X_nX_n^T$ is a Wishart Matrix whose eigenvalues will follow, in the limit, the Marchenko-Pastur distribution. Howerever, I am interested in finite $n$ results.

Computer Simulations suggest that $\alpha_n$ increases with $n$. I suspect this might be a known fact that I can't seem to find in the literature, I would really appreciate if someone could point me to the right reference!

Answer 1

You ask for the average of the singular values of the Wishart matrix. I'm pretty sure there is no closed form expression valid for any $n$. If instead you would ask for the average of the square of the singular value, then the answer is very simple, this is just unity independent of $n$.

More generally, if $X_{n}$ has dimension $p\times n$, with $p\leq n$, and $w_k$ is an eigenvalue of $W_n=n^{-1}X_nX_n^T$, then

$$E\left(p^{-1}\sum_{k=1}^{p}w_k\right)=1,$$

regardless of the value of $p$ or $n$. This follows directly from equation 17.8.2 in Mehta's book on Random-Matrix Theory, applied to the Wishart-Laguerre probability distribution of the $w_k$'s.

Similar closed form expressions exist for the integer moments $E(p^{-1}\sum_k w_k^m)$, see this paper by Livan and Vivo. The singular values are a fractional moment ($m=1/2$). In principle these are given by an integral over Laguerre polynomials, but I do not think this integral can be carried out in closed form.

Answer 2

I don't have time for details right now, but you want to look at circa 1990 papers by Alan Edelman and Stanislaw Szarek (independently) on condition numbers of random matrices. For the state of the art (as of 2009) see this blog post by Terry Tao.