448 research outputs found

    On the Law of Addition of Random Matrices

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    Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices AnA_{n} and BnB_{n} rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix UnU_{n} (i.e. An+Unβˆ—BnUnA_{n}+U_{n}^{\ast}B_{n}U_{n}) is studied in the limit of large matrix order nn. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of AnA_{n} and BnB_{n} is obtained and studied. Keywords: random matrices, eigenvalue distributionComment: 41 pages, submitted to Commun. Math. Phy

    Central limit theorem for linear eigenvalue statistics of random matrices with independent entries

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    We consider nΓ—nn\times n real symmetric and Hermitian Wigner random matrices nβˆ’1/2Wn^{-1/2}W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices nβˆ’1Xβˆ—Xn^{-1}X^*X with independent entries of mΓ—nm\times n matrix XX. Assuming first that the 4th cumulant (excess) ΞΊ4\kappa_4 of entries of WW and XX 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 nβ†’βˆžn\to\infty, mβ†’βˆžm\to\infty, m/nβ†’c∈[0,∞)m/n\to c\in[0,\infty) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C5\mathbf{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 C5\mathbb{C}^5 test function. Here the variance of statistics contains an additional term proportional to ΞΊ4\kappa_4. The proofs of all limit theorems follow essentially the same scheme.Comment: Published in at http://dx.doi.org/10.1214/09-AOP452 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
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