Related papers: Gaussian Fluctuations for Sample Covariance Matric…
Taylor's power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. It has been shown to hold by empirical observations in a…
We prove the Marchenko-Pastur law for the eigenvalues of $p \times p$ sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario - the block-independent model - the $p$…
We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let $X_n$ be a Hermitian random matrix of size $n\times n$ that can be split into independent blocks…
We prove a local law in the bulk of the spectrum for random Gram matrices $XX^*$, a generalization of sample covariance matrices, where $X$ is a large matrix with independent, centered entries with arbitrary variances. The limiting…
For a generalization of Johnstone's spiked model, a covariance matrix with eigenvalues all one but $M$ of them, the number of features $N$ comparable to the number of samples $n: N=N(n), M=M(n), \gamma^{-1} \leq \frac{N}{n} \leq \gamma$…
Consider the $n \times n$ reverse circulant $RC_n(t)$ and symmetric circulant $SC_n(t)$ matrices with independent Brownian motion entries. We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics…
Consider the product $X = X_{1}\cdots X_{m}$ of $m$ independent $n\times n$ iid random matrices. When $m$ is fixed and the dimension $n$ tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of $X$ for…
This paper investigates the rate of convergence for the central limit theorem of linear spectral statistic (LSS) associated with large-dimensional sample covariance matrices. We consider matrices of the form ${\mathbf…
In this article, we study the fluctuation of linear eigenvalue statistics of symmetric circulant matrices $(SC_n)$ with independent entries which satisfy some moment conditions. We show that $\frac{1}{\sqrt{n}} \Tr \phi(SC_n)$ obey the…
Several well-known results from the random matrix theory, such as Wigner's law and the Marchenko--Pastur law, can be interpreted (and proved) in terms of non-backtracking walks on a certain graph. Orthogonal polynomials with respect to the…
In this article we study the fluctuation of linear statistics of eigenvalues of circulant, symmetric circulant, reverse circulant and Hankel matrices. We show that the linear spectral statistics of these matrices converges to the Gaussian…
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…
Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real $p \times n$ matrix with independent entries. It is well known that if the entries of $X$ are independent and identically distributed (i.i.d.) with enough moments…
Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the…
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…
For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a…
This paper is concerned with the limiting spectral behaviors of large dimensional Kendall's rank correlation matrices generated by samples with independent and continuous components. We do not require the components to be identically…
Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded…
We show that the global fluctuations of spectra of GOE and GUE matrices and their principal submatrices executing Dyson's Brownian motion are Gaussian in the limit of large matrix dimensions. For nested submatrices one obtains a limiting…
In random matrix theory, Marchenko-Pastur law states that random matrices with independent and identically distributed entries have a universal asymptotic eigenvalue distribution under large dimension limit, regardless of the choice of…