459 research outputs found
On the Law of Addition of Random Matrices
Normalized eigenvalue counting measure of the sum of two Hermitian (or real
symmetric) matrices and rotated independently with respect to
each other by the random unitary (or orthogonal) Haar distributed matrix
(i.e. ) is studied in the limit of large
matrix order . 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 and 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
We consider real symmetric and Hermitian Wigner random matrices
with independent (modulo symmetry condition) entries and the (null)
sample covariance matrices with independent entries of
matrix . Assuming first that the 4th cumulant (excess) of entries
of and 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 , , with the same variance as for Gaussian matrices if the test
functions of statistics are smooth enough (essentially of the class
). 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 test function.
Here the variance of statistics contains an additional term proportional to
. 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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