25 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 Non-Gaussian Limiting Laws for Certain Statistics of Wigner Matrices
This paper is a continuation of our papers [12-14] in which the limiting laws of fluctuations were found for the linear eigenvalue statistics Tr φ(M⁽ⁿ⁾) and for the normalized matrix elements √nφjj(M⁽ⁿ⁾) of differentiable functions of real symmetric Wigner matrices M⁽ⁿ⁾ as n →∞.Статья является продолжением исследования, начатого в работах [12-14],где были найдены предельные законы флуктуаций для линейных статистик собственных значений Tr φ(M⁽ⁿ⁾) и нормированных матричных элементов √nφjj(M⁽ⁿ⁾) дифференцируемых функций от вещественных симметричных матриц Вигнера M⁽ⁿ⁾ при n →∞
Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices
We study the fluctuations of the matrix entries of regular functions of
Wigner random matrices in the limit when the matrix size goes to infinity. In
the case of the Gaussian ensembles (GOE and GUE) this problem was considered by
A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are
valid provided the off-diagonal matrix entries have finite fourth moment, the
diagonal matrix entries have finite second moment, and the test functions have
four continuous derivatives in a neighborhood of the support of the Wigner
semicircle law.Comment: minor corrections; the manuscript will appear in the Journal of
Statistical Physic
On universality of local edge regime for the deformed Gaussian Unitary Ensemble
We consider the deformed Gaussian ensemble in which
is a hermitian matrix (possibly random) and is the Gaussian
unitary random matrix (GUE) independent of . Assuming that the
Normalized Counting Measure of converges weakly (in probability if
random) to a non-random measure with a bounded support and assuming
some conditions on the convergence rate, we prove universality of the local
eigenvalue statistics near the edge of the limiting spectrum of .Comment: 25 pages, 2 figure
Functional limit theorems for random regular graphs
Consider d uniformly random permutation matrices on n labels. Consider the
sum of these matrices along with their transposes. The total can be interpreted
as the adjacency matrix of a random regular graph of degree 2d on n vertices.
We consider limit theorems for various combinatorial and analytical properties
of this graph (or the matrix) as n grows to infinity, either when d is kept
fixed or grows slowly with n. In a suitable weak convergence framework, we
prove that the (finite but growing in length) sequences of the number of short
cycles and of cyclically non-backtracking walks converge to distributional
limits. We estimate the total variation distance from the limit using Stein's
method. As an application of these results we derive limits of linear
functionals of the eigenvalues of the adjacency matrix. A key step in this
latter derivation is an extension of the Kahn-Szemer\'edi argument for
estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and
Related Field