448 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 $A_{n}$ and $B_{n}$ rotated independently with respect to
each other by the random unitary (or orthogonal) Haar distributed matrix
$U_{n}$ (i.e. $A_{n}+U_{n}^{\ast}B_{n}U_{n}$) is studied in the limit of large
matrix order $n$. 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 $A_{n}$ and $B_{n}$ 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

We consider $n\times n$ 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 $m\times n$
matrix $X$. Assuming first that the 4th cumulant (excess) $\kappa_4$ 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 $n\to\infty$, $m\to\infty$, $m/n\to
c\in[0,\infty)$ with the same variance as for Gaussian matrices if the test
functions of statistics are smooth enough (essentially of the class
$\mathbf{C}^5$). 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 $\mathbb{C}^5$ test function.
Here the variance of statistics contains an additional term proportional to
$\kappa_4$. 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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