Tracy - Widom law for the extreme eigenvalues of sample correlation matrices ∗

Let the sample correlation matrix be W = Y Y T , where Y = (yij)p,n with yij = xij/ √∑n j=1 x 2 ij . We assume {xij : 1 ≤ i ≤ p, 1 ≤ j ≤ n} to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any i, we assume xij , 1 ≤ j ≤ n to be identically distributed. We assume 0 < p < n and p/n → y with some y ∈ (0, 1) as p, n → ∞. In this paper, we provide the Tracy-Widom law (TW1) for both the largest and smallest eigenvalues of W . If xij are i.i.d. standard normal, we can derive the TW1 for both the largest and smallest eigenvalues of the matrix R = RR , where R = (rij)p,n with rij = (xij − x̄i)/ √∑n j=1(xij − x̄i), x̄i = n −1 ∑n j=1 xij .