Please use this identifier to cite or link to this item: https://hdl.handle.net/10356/96096
Title: Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
Authors: Bao, Zhigang
Pan, Guangming
Zhou, Wang
Issue Date: 2012
Source: Bao, Z., Pan, G., & Zhou, W. (2012). Tracy-Widom law for the extreme eigenvalues of sample correlation matrices. Electronic Journal of Probability, 17(88), 1-32.
Series/Report no.: Electronic journal of probability
Abstract: 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.
URI: https://hdl.handle.net/10356/96096
http://hdl.handle.net/10220/10085
ISSN: 1083-6489
DOI: http://dx.doi.org/10.1214/EJP.v17-1962
Rights: © 2012 The Authors. This paper was published in Electronic Journal of Probability and is made available as an electronic reprint (preprint) with permission of The Authors. The paper can be found at the following official DOI: [http://dx.doi.org/10.1214/EJP.v17-1962].  One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.
metadata.item.grantfulltext: open
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