Edge universality of correlation matrices
By Natesh S. Pillai, Jun Yin. 2011
51 citations recorded by the source corpus.
xij = 0 and E e 2 = 1 M , N,M → ∞. These entries have a sub-exponential decay at the tails. We will be working in the regime N/M = dN,limN!1 dN 6 0,1,∞. In this paper we prove the edge universality of correlation matrices X † X, where the rectangular matrix X (called the standardized matrix) is obtained by normalizing each column of the data matrix e X by its Euclidean norm. Our main result states that, asymptotically the k-point (k > 1) correlation functions of the extreme eigenvalues (at both edges of the spectrum) of the correlation matrix X † X converge to those of the Gaussian correlation matrix, i.e., Tracy-Widom law, and thus in particular, the largest and the smallest eigenvalues of X † X after appropriate centering and rescaling converge to the Tracy-Widom distribution. The asymptotic distribution of extreme eigenvalues of the Gaussian correlation matrix has been worked out only recently. As a corollary of the main result in this paper, we also obtain that the extreme eigenvalues of Gaussian correlation matrices are asymptotically distributed according to the TracyWidom law. The proof is based on the comparison of Green functions, but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix. We achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix. Our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/dfe2081d9e4c2ddddebb589857f34e1817b73b78
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In Discrete Mathematics and Combinatorics
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Citations in the corpus
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “was raised in the series of work [21–23] on the local eigenvalue statistics of Wigner matrices originally, and has shown its strong applicability on some other random matrix models or statistics; see [5, 15, 45] for its variants for sample covariance and correlation matrices and see [50] for an appl”
- A necessary and sufficient condition for edge universality of Wigner matrices (2012, methodology): “α = 4 in (1.3) will provide the marginal case. The edge universalityhasbeen generalizedin manydirections, forexample, forthe sample covariance matrices [28, 31, 46, 49, 41] and for correlation matrix [4, 42]. For the deformed matrices, which are described as a finite rank perturbation of sample covar”
- Random matrix theory in statistics: A review (2014, background): “Universality at the edge of the spectrum of sample covariance and correlation matrices has been studied by Feldheim and Sodin (2010), Bao et al. (2012) and Pillai and Jin (2012).”
- A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, methodology): “This goal can be accomplished with a Green function comparison method, which has been applied successfully in proving the universality of covariance matrices [29, 30].”
- High dimensional correlation matrices: the central limit theorem and its applications (2017, background): “In addition, both Bao et al. (2012) and Pillai and Yin (2012) established asymptotic distributions for the extreme eigenvalues of the sample correlation matrices under study.”
- Universality of covariance matrices (2011, methodology): “For instance, the matrix X = [xij ] constructed above is called the correlation matrix (see [22, 33]) and is often preferred in applications such as principal component analysis (PCA) due to the scale invariance of the correlation matrix.”