arXiv subjects

V. Kargin

Publications and source records attributed to V. Kargin.

3 recordsLinked to original sources

arXiv2015

Subordination for the sum of two random matrices

This paper is about the relation of random matrix theory and the subordination phenomenon in complex analysis. We find that the resolvent of the sum of two random matrices is approximately subordinated to the resolvents of the original matrices. We estimate the error terms in this relation and in the subordination relation for the traces of the resolvents. This allows us to prove a local limit law for eigenvalues and a delocalization result for eigenvectors of the sum of two random matrices. In addition, we use subordination to determine the limit of the largest eigenvalue for the rank-one deformations of unitary-invariant random matrices.

math.PR↗

arXiv2013

An inequality for the distance between densities of free convolutions

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $μ_i$ and $ν_i, i=1,2$, are close to each other in terms of the Lévy metric and if the free convolution $μ_1\boxplusμ_2$ is sufficiently smooth, then $ν_1\boxplusν_2$ is absolutely continuous, and the densities of measures $ν_1\boxplusν_2$ and $μ_1\boxplusμ_2$ are close to each other. In particular, convergence in distribution $μ_1^{(n)}\rightarrow μ_1,$ $μ_2^{(n)}\rightarrowμ_2$ implies that the density of $μ_1^{(n)}\boxplusμ_2^{(n)}$ is defined for all sufficiently large $n$ and converges to the density of $μ_1\boxplusμ_2$. Some applications are provided, including: (i) a new proof of the local version of the free central limit theorem, and (ii) new local limit theorems for sums of free projections, for sums of $\boxplus$-stable random variables and for eigenvalues of a sum of two $N$-by-$N$ random matrices.

math.PR↗

arXiv2009

Free point processes and free extreme values

We continue here the study of free extreme values begun in Ben Arous and Voiculescu (2006). We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure (Voiculescu (1998), Barndorff-Nielsen and Thorbjornsen (2005)). We relate this convergence to the free extremal laws introduced in Ben Arous and Voiculescu (2006) and give the limit laws for free order statistics.

math.PR↗