A Universality Result for the Smallest Eigenvalues of Certain Sample Covariance Matrices

2008

153 citations recorded by the source corpus.

After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy–Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Péché.

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

View Paper: https://www.semanticscholar.org/paper/1f2e6ffb158fed8427915612d9dd68cc3f87b14a

Citations in the corpus

• Universality of covariance matrices (2011, unknown): “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”
• Small deviations for beta ensembles (2009, unknown): “This appraisal was long understood for the minimal eigenvalue of L{O/U}E, and has recently been extended to non-Gaussian versions of those ensembles in [8].”
• Random matrix theory in statistics: A review (2014, unknown): “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).”
• Non-asymptotic theory of random matrices: extreme singular values (2010, unknown): “The method of [25] also addresses the more difficult smallest singular value.”
• The local semicircle law for a general class of random matrices (2012, unknown): “16) is fairly involved; it makes use of the Chebyshev polynomial representation first used by Feldheim and Sodin [17,22] in this context for a special distribution of hij , and extended in [2] to general symmetric entries.”
• Random covariance matrices: Universality of local statistics of eigenvalues (2009, unknown): “For some earlier results in this direction, we refer to [2, 13, 25, 26] and the references therein.”
• Sample covariance matrices of heavy-tailed distributions (2016, unknown): “lassical works [33] and [3] regarding almost sure convergence of appropriately normalized singular values when the coordinates of the underlying distributions are i.i.d.; as well as more recent works [22, 14, 18, 25, 9, 1, 2, 29, 19, 26, 20, 17, 28, 11, 10, 31, 32, 21, 27, 8]. For a more comprehensi”
• RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES UP TO THE EDGE (2011, unknown): “Later, Feldheim and Sodin [8] proved a universality result for certain sample covariance matrices: Theorem 2.”
• Maximum likelihood estimation for linear Gaussian covariance models (2014, unknown): “In fact, this convergence rate holds in more general scenarios with the standard normalization without logarithms [FS10].”
• Delocalization and Diffusion Profile for Random Band Matrices (2012, unknown): “The work [6, 7] was partly motivated by a similar results for the long-time evolution for the Anderson model [15–17] combined with an algebraic renormalization using Chebyshev polynomials [26,34].”
• A necessary and sufficient condition for edge universality of Wigner matrices (2012, unknown): “cal simulation results in [7] also suggest that 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 descri”
• The local relaxation flow approach to universality of the local statistics for random matrices (2009, unknown): “First, we recall the following result from [22].”
• Spectral norm of products of random and deterministic matrices (2008, unknown): “1”
• Random Matrices: the Distribution of the Smallest Singular Values (2009, unknown): “(44) Let F be a function as in the statement of the theorem.”
• Edge universality of correlation matrices (2011, unknown): “In [14, 23] and [25], Peche, Soshnikov and Sodin proved that for some covariance matrices (including the Wishart matrix), the largest and smallest k eigenvalues after appropriate centering and rescaling converge in distribution to the Tracy–Widom law4 whose density is a smooth function.”
• Sharp nonasymptotic bounds on the norm of random matrices with independent entries (2014, unknown): “ combinatorial approach would be nontrivial as very high moments of Wigner matrices appear in this argument; however, the ingredients that would be necessary for a combinatorial proof can be found in [8].) We conclude the introduction by noting that both the noncommutative Khintchine inequality and ”