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On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity

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arxiv math/0309355 v1 pith:FWEIJ46H submitted 2003-09-22 math.ST stat.TH

Noureddine El Karoui This is my paper · ORCID
classification math.STstat.TH
keywords wheninfinitylargestcovarianceeigenvalueevenjohnstonematrix
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Let X be a n*p matrix and l_1 the largest eigenvalue of the covariance matrix X^{*}*X. The "null case" where X_{i,j} are independent Normal(0,1) is of particular interest for principal component analysis. For this model, when n, p tend to infinity and n/p tends to gamma in (0,\infty), it was shown in Johnstone (2001) that l_1, properly centered and scaled, converges to the Tracy-Widom law. We show that with the same centering and scaling, the result is true even when p/n or n/p tends to infinity. The derivation uses ideas and techniques quite similar to the ones presented in Johnstone (2001). Following Soshnikov (2002), we also show that the same is true for the joint distribution of the k largest eigenvalues, where k is a fixed integer. Numerical experiments illustrate the fact that the Tracy-Widom approximation is reasonable even when one of the dimension is "small".

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Cited by 4 Pith papers

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  1. Ties, Tails and Spectra: On Rank-Based Dependency Measures in High Dimensions

    math.ST 2025-08 conditional novelty 7.0 of 10

    Spearman's rho and a rescaled Kendall's tau matrices have universal limiting spectral distributions (semicircle and Marchenko-Pastur) for high-dimensional data with ties or heavy tails.

  2. Universal Bootstrap for Spectral Statistics: Beyond Gaussian Approximation

    math.ST 2024-12 conditional novelty 7.0 of 10

    A Gaussian-replacement universal bootstrap is shown to be consistent for operator-norm spectral statistics when p/n is bounded or diverges to infinity, with no eigenvalue-decay assumptions.

  3. Asymptotic Expansions of the Limit Laws of Gaussian and Laguerre (Wishart) Ensembles at the Soft Edge

    math.PR 2024-03 unverdicted novelty 7.0 of 10

    Explicit asymptotic expansions in powers of h ~ n^{-2/3} are derived for the Tracy-Widom distributions F_beta describing the rescaled largest eigenvalues of Gaussian and Laguerre ensembles, with polynomial coefficient...

  4. Application of Random Matrix Theory in High-Dimensional Statistics

    stat.ME 2024-12 conditional novelty 3.0 of 10

    A review of RMT in high-dimensional statistics that contributes a new CLT for the log-eigenvalues of Wishart matrices, with a flawed proof.

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