Zhigang Bao

Local Law of Addition of Random Matrices on Optimal Scale

The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.

SPECTRAL STATISTICS OF LARGE DIMENSIONAL SPEARMAN'S RANK CORRELATION MATRIX AND ITS APPLICATION

between two independent random variables. In this paper we will establish a CLT for the linear spectral statistics of this non-parametric random matrix model in the scenario of high dimension supposing that p = p(n) and p=n! c2 (0;1) as n!1. We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni’s cumulant method in [1] to bypass the so called joint cumulant summability. In addition, we raise a two-step comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT we then construct a distribution-free statistic to test complete independence for components of random vectors. Owing to the non-parametric property, we can use this test on generally distributed random variables including the heavy-tailed ones.

Test of independence for high-dimensional random vectors based on freeness in block correlation matrices

In this paper, we are concerned with the independence test for k high-dimensional sub-vectors of a normal vector, with fixed positive integer k. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is proposed for this purpose. We then construct the so-called Schott type statistic as our test statistic, which turns out to be a particular linear spectral statistic of the block correlation matrix. Interestingly, the limiting behavior of the Schott type statistic can be figured out with the aid of the Free Probability Theory and the Random Matrix Theory. Specifically, we will bring the so-called real second order freeness for Haar distributed orthogonal matrices, derived in Mingo and Popa (2013)[10], into the framework of this high-dimensional testing ∗Corresponding author. †Z.G. Bao was supported by a startup fund from HKUST. ‡J. Hu was partially supported by Science and Technology Development Foundation of Jilin (Grant No. 20160520174JH), Science and Technology Foundation of Jilin during the “13th Five-Year Plan” and the National Natural Science Foundation of China (Grant No. 11301063). §G.M. Pan was partially supported by a MOE Tier 2 grant 2014-T2-2-060 and by a MOE Tier 1 Grant RG25/14 at the Nanyang Technological University, Singapore. ¶W. Zhou was partially supported by R-155-000-165-112 at the National University of Singapore. 1527

The logarithmic law of random determinant

Consider the square random matrix

Universality for a global property of the eigenvectors of Wigner matrices

A_n=(a_{ij})_{n,n}

Local Semicircle law and Gaussian fluctuation for Hermite

, where

\beta

\{a_{ij}:=a_{ij}^{(n)},i,j=1,\ldots,n\}

ensemble

is a collection of independent real random variables with means zero and variances one. Under the additional moment condition \[\sup_n\max_{1\leq i,j\leq n}\mathbb{E}a_{ij}^4<\infty,\] we prove Girkos logarithmic law of

CLT for Linear Spectral Statistics of Hermitian Wigner Matrices with General Moment Conditions

\det A_n

Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices

in the sense that as