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

Abstract

Let the sample correlation matrix be W = YY Twhere Y = (y ij) p;n with y ij. We assume to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any i, we assume x ij, 1 ≤ j ≤ n to be identically distributed. We assume 0 < p < n and p=n → y with some y ε (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 x ij are i.i.d. standard normal, we can derive the TW 1 for both the largest and smallest eigenvalues of the matrix R = RR T, where R = (r ij) p;n with r ij.