Related papers: Linear eigenvalue statistics of random matrices wi…
We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. In particular, we follow the methodology…
We study the real eigenvalue statistics of products of independent real Ginibre random matrices. These are matrices all of whose entries are real i.i.d. standard Gaussian random variables. For such product ensembles, we demonstrate the…
Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,...,$ being independent real random variables such that \be \mathbb{E}[a_{j}]=0, \ \ \mathbb{E}[|a_{j}|^{2}]=1 \ \ \textrm{for}\,\…
We consider large non-Hermitian random matrices $X$ with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having…
This work considers the problem of estimating the distance between two covariance matrices directly from the data. Particularly, we are interested in the family of distances that can be expressed as sums of traces of functions that are…
We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of measures. The limit is (not the normal distribution and is)…
This article focuses on linear eigenvalue statistics of Hankel matrices with independent entries. Using the convergence of moments we show that the linear eigenvalue statistics of Hankel matrices for odd degree monomials with degree greater…
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…
In Part I of this article (Banerjee and Kuchibhotla (2023)), we have introduced a new method to bound the difference in expectations of an average of independent random vector and the limiting Gaussian random vector using level sets. In the…
We study the number of occurrences of any fixed vincular permutation pattern. We show that this statistics on uniform random permutations is asymptotically normal and describe the speed of convergence. To prove this central limit theorem,…
We consider $N$ by $N$ deformed Wigner random matrices of the form $X_N=H_N+A_N$, where $H_N$ is a real symmetric or complex Hermitian Wigner matrix and $A_N$ is a deterministic real bounded diagonal matrix. We prove a universal Central…
We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large $n$ for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form…
A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of…
In this paper we consider the product of two independent random matrices $\mathbb X^{(1)}$ and $\mathbb X^{(2)}$. Assume that $X_{jk}^{(q)}, 1 \le j,k \le n, q = 1, 2,$ are i.i.d. random variables with $\mathbb E X_{jk}^{(q)} = 0, \mathbb E…
In this paper, we establish some new central limit theorems for certain spectral statistics of a high-dimensional sample covariance matrix under a divergent spectral norm population model. This model covers the divergent spiked population…
In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we…
In the first part of these notes, we review some of the recent developments in the study of the spectral properties of Wigner matrices. In the second part, we present a new proof of a Wegner estimate for the eigenvalues of a large class of…
This paper investigates a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors, that is,…
We study the fluctuations of the eigenvalues of real valued large centrosymmetric random matrices via its linear eigenvalue statistic. This is essentially a central limit theorem (CLT) for sums of dependent random variables. The dependence…
We show how positive unital linear maps can be used to obtain lower bounds for the maximum distance between the eigenvalues of two normal matrices. Some related bounds for the spread and condition number of Hermitian matrices are also…