58 research outputs found
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
On a Limiting Distribution of Singular Values of Random Band Matrices
An equation is obtained for the Stieltjes transform of the normalized
distribution of singular values of non-symmetric band random matrices in the
limit when the band width and rank of the matrix simultaneously tend to
infinity. Conditions under which this limit agrees with the quarter-circle law
are found. An interesting particular case of lower triangular random matrices
is also considered and certain properties of the corresponding limiting
singular value distribution are given
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