Universality of the top eigenvalue of correlation matrices

Question

Let $X$ be a $N\times P$ matrix with random independent and identically distributed entries $x_{ij}$. I also assume that $\langle x\rangle = 0$ and $\langle x^2\rangle = 1$.

Define the $N\times N$ matrix $C = (1/N)XX^T$.

I am interested in the limit $N\rightarrow\infty$, with $P=\alpha N$ for some finite positive constant $\alpha$. In this limit, the top eigenvalue of $C$ concentrates around a value. If the entries of $X$ are standard normal, then

$$\lambda_\mathrm{max} = (1 + \sqrt\alpha)^2$$

as known from the Marcenko and Pastur result.

I found a paper that claims that the same result is valid for any distribution of the $x_{ij}$, with the same first two moments. But this result is not proved, but only stated in passing. If this is well known, can someone provide a reference or explanation of why this is the case?

Answer 1

In general you need more than second moment - you need fourth moment finite, otherwise the top eigenvalue can run off to infinity in your scaling. For this and more see the book of Bai and Silverstein (and the original papers).