# Universality of the top eigenvalue of correlation matrices

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?

• This is done in detail in the book of Bai and Silverstein. It is pretty standard - either by the moment method (after truncating) or computing Stieltjes transforms – ofer zeitouni Mar 9 '20 at 17:35
• @oferzeitouni Thanks! The result is stronger: the full spectrum (not just its extremes) is universal. For future reference, this is theorem 3.7 in the Bail & Silverstein book. – becko Mar 9 '20 at 17:57
• If you post an answer I'll accept it @oferzeitouni – becko Mar 9 '20 at 17:57
• For the extreme you need more. You need 4th moment. Otherwise the top eigenvalue can run off to infinity in the standard scaling (I misread your question - thought you cared only about the convergence of empirical measure) – ofer zeitouni Mar 9 '20 at 18:28