CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES

By Anna Lytova, Leonid A. Pastur. 2008

284 citations recorded by the source corpus.

We consider n x n real symmetric and Hermitian Wigner random matrices n ―1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ―1 X*X with independent entries of m x n matrix X. Assuming first that the 4th cumulant (excess) κ 4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, oo) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5 ). This is done by using a simple "interpolation trick" from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially ℂ 5 test function. Here the variance of statistics contains an additional term proportional to κ 4 . The proofs of all limit theorems follow essentially the same scheme.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f8fd7dbd754fbc6563ca80385e2ae97557c6569e

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Citations in the corpus

Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing (2014, background): “This LSD is referred as the generalized Marčenko-Pastur distribution with index (y,H) (for background on MarčenkoPastur distributions, the reader is referred to Bai and Silverstein [6, Chapter 3]).”
Random matrix theory in statistics: A review (2014, methodology): “…proving the results for matrices with Gaussian entries and then utilizing an interpolation between the random matrix ensemble of interest and a conveniently chosen Gaussian matrix, and then making use of an extension of the well-known Stein's Lemma (see Proposition 3.1 of Lytova and Pastur, 2009).”
Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices (2013, background): “…via a potential in [32], the set of test functions was extended and the assumptions on the entries of the Wigner matrices weakened in [7, 6, 36, 42], a more general model of band matrices was considered in [2] (see also [36, 5] for general covariance matrices), unitary matrices where…”
Anisotropic local laws for random matrices (2014, background): “60026 [28] Lytova, A; Pastur, L, Central limit theorem for linear eigenvalue statistics of random matrices with independent entries, Ann.”
Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices (2012, background): “Pastur [26] have proved that for GOE matrices and φ ∈ C1(R) satisfying suitable decay conditions at infinity, N ◦ n [φ] converges to a Gaussian random variable with mean zero and variance”
GLOBAL FLUCTUATIONS FOR LINEAR STATISTICS OF β-JACOBI ENSEMBLES (2012, methodology): “We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.”
Statistical inference for principal components of spiked covariance matrices (2020, unknown): “[54] A.”
Universality and Sharp Matrix Concentration Inequalities (2022, methodology): “In contrast, prior universality results such as those in [17, 13] are limited to the “classical random matrix regime” where the entry variances are of order d− 1 2 and exhibit decay of correlations.”
Corrections to LRT on large-dimensional covariance matrix by RMT (2009, background): “In the “white” case, Tn ≡ I as considered here, in a recent preprint Pastur and Lytova [7], the authors offer a new extension of the CLT where the constraints E|ξ11|4 = 3 or 2, as stated above, are removed.”
Independence test for high dimensional data based on regularized canonical correlation coefficients (2015, methodology): “To overcome this difficulty, we employ an interpolation trick first adopted in Lytova and Pastur (2009) and extend the result to the nonGaussian distributions.”