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?