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

• Main Authors: Bao, Zhigang, Pan, Guangming, Zhou, Wang
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2013
• Online Access: https://hdl.handle.net/10356/96096; http://hdl.handle.net/10220/10085
• Institution: Nanyang Technological University

Let the sample correlation matrix be W = YYT, where Y = (yij)p,n with yij = xij /√∑xij2. 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 = RRT, where R = (rij)p,n with rij = (xij − xi )/√∑(xij −xi)2, xi = n−1∑xij.