It is well known that an @math Wishart matrix with @math degrees of freedom is close to the appropriately centered and scaled Gaussian Orthogonal Ensemble (GOE) if @math is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when @math . Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when @math . This shows, in particular, that the phase transition from Wishart to GOE is smooth.
Theorems and state that as @math , statistics of the Wishart @math and the GOE @math have asymptotically the same distribution, but this is not the case if @math remains bounded. In the random matrix literature there has been lots of work showing that statistics of these ensembles have asymptotically the same distribution even when @math . For instance, when @math , then the limiting empirical spectral distribution of the Wishart is the Marchenko-Pastur law, which shows the difference between the Wishart and GOE, but the largest eigenvalue of the Wishart already behaves like that of the GOE @cite_2 @cite_8 @cite_6 . This naturally raises the question of whether there are other regimes of @math and @math where there are interesting phase transitions.
@cite_8 @cite_6 @cite_2
2079985400 1520752838
We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n x p matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance Σ p . We show that for a large class of covariance matrices £ p, the largest eigenvalue of X*X is asymptotically distributed (after recentering and rescaling) as the Tracy-Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, n and p. The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics. Let x (1) denote the square of the largest singular value of an n x p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component variance of the covariance matrix X'X, or the largest eigenvalue of a p-variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n p = y ≥ 1. When centered by μ p = (√n-1 + √p) 2 and scaled by σ p = (√n-1 + √p)(1 √n-1 + 1 √p) 1 3 , the distribution of x (1) approaches the Tracy-Widom law of order 1, which is defined in terms of the Painleve II differential equation and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.