A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices

By Alexander Soshnikov. 2001

246 citations recorded by the source corpus.

Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X*X(XtX) converges to the Tracy–Widom law as n,p (the dimensions of X) tend to ∞ in some ratio n/p→γ>0. We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n−p=O(p1/3). We also prove that λmax≤(n1/2+p1/2)2+O(p1/2 log(p)) (a.e.) for general γ>0.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

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Citations in the corpus

RUDDER: Return Decomposition for Delayed Rewards (2018, methodology): “ular matrices says that for n= mthe maximal singular value is 2 p p m[79]. Asymptotically we have for the maximal singular value smax(A) / m+ n[104]. A bound on the largest singular value is given by [122]: s2 max(A) 6 ( p m+ p n)2 + O( p nlog(n)) a.s. (A236) Therefore, a rectangular matrix modiﬁes ”
Introduction to the non-asymptotic analysis of random matrices (2010, background): “Moreover, the limiting distribution of smax(A) is known to be the Tracy-Widom law (see [71, 27]).”
On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity (2003, methodology): “en Johnstone’s considerations hold true all the way and we have the same conclusion as in Theorem 1. We note that using the interlacing properties of the singular values (as mentioned for instance in Soshnikov (2002), Remark 5; see also Horn and Johnson (1990), theorem 7.3.9), as well as the estimat”
Universality of covariance matrices (2011, methodology): “In [34], [31] and [19], Soshnikov, Péché, Feldheim and Sodin proved that for the covariance matrices whose entries have a symmetric probability density function (which includes the Wishart matrix), the largest and smallest k eigenvalues after appropriate centering and rescaling converge in distribut”
Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “The readers can refer to [43, 46, 49, 56] for some representative developments on this topic.”
Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices (2012, unknown): “The strategy follows from [30], Remark 5.”
On the principal components of sample covariance matrices (2014, background): “independent of the distribution of the entries of X, in [36, 40].”
The Surprising Mathematics of Longest Increasing Subsequences (2015, background): “This was proved in various levels of generality in [97], [99], [122], [144].”
On Spectral Learning of Mixtures of Distributions (2005, methodology): “Using techniques from Soshnikov [7] describing concentration of the median of σ for various sample counts, one can show that a d -dimensional Gaussian with maximum directional variance σ 2 is g -convergent for g ( δ ) = cd log(1 /δ ) for a universal constant c .”
The Littlewood-Offord problem and invertibility of random matrices (2007, background): “Indeed, Soshnikov [25] proved that the limiting distribution of s1(A) is precisely the Tracy– Widom law for all matrices with i.”
A necessary and sufficient condition for edge universality of Wigner matrices (2012, background): “The edge universality has been generalized in many directions, for example, for the sample covariance matrices [28, 31, 46, 49, 41] and for correlation matrix [4, 42].”
Edge universality of correlation matrices (2011, methodology): “Here we use the term Tracy–Widom law as in [25].”
Non-asymptotic theory of random matrices: extreme singular values (2010, background): “For the largest singular values of random matrices with independent entries it was established by Johansson [37] and Johnstone [39] in the Gaussian case, and by Soshnihikov [74] for more general distributions.”
Universality results for largest eigenvalues of some sample covariance matrix ensembles (2007, methodology): “These numbers are the major combinatorial tools needed to adapt the computations of [30] to sample covariance matrices of any sample size to dimension ratio γN .”
THE SPECTRUM OF RANDOM INNER-PRODUCT KERNEL MATRICES (2012, unknown): “Singer/Random Kernel Matrices 29 [20] Soshnikov, A.”
TESTING HYPOTHESES ABOUT THE NUMBER OF FACTORS IN LARGE FACTOR MODELS (2009, background): “The first important universality results were recently obtained by Soshnikov (2002) .”
Principal components analysis corrects for stratification in genome-wide association studies (2006, methodology): “, unpublished data), focusing instead on individual genotype data and placing the method on a firm statistical footing by rigorously assigning statistical significance to each axis of variatio”