Related papers: Random matrix products: Universality and least sin…

We consider real, Gauss-divisible matrices $A_{t}=A+\sqrt{t}B$, where $B$ is from the real Ginibre ensemble. We prove that the bulk correlation functions converge to a universal limit for $t=O(N^{-1/3+\epsilon})$ if $A$ satisfies certain…

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random $N\times N$ matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an infinite…

In statistics, assuming samples are independent is reasonable. However, this property can fail to hold for the features, a distinction that has led to several lines of work aiming to remove the latter assumption of independence present in…

We consider the following data perturbation model, where the covariates incur multiplicative errors. For two $n \times m$ random matrices $U, X$, we denote by $U \circ X$ the Hadamard or Schur product, which is defined as $(U \circ X)_{ij}…

Random linear mappings are widely used in modern signal processing, compressed sensing and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful…

Consider the product of $m$ independent $n\times n$ random matrices from the spherical ensemble for $m\ge 1$. The empirical distribution based on the $n$ eigenvalues of the product is called the empirical spectral distribution. Two recent…

We consider products of independent large random rectangular matrices with independent entries. The limit distribution of the expected empirical distribution of singular values of such products is computed. The distribution function is…

Consider fixed and bounded trace Gaussian orthogonal, unitary and symplectic ensembles, closely related to Gaussian ensembles without any constraint. For three restricted trace Gaussian ensembles, we prove universal limits of correlation…

Sums of independent random variables form the basis of many fundamental theorems in probability theory and statistics, and therefore, are well understood. The related problem of characterizing products of independent random variables seems…

We propose a novel approach to concentration for non-independent random variables. The main idea is to ``pretend'' that the random variables are independent and pay a multiplicative price measuring how far they are from actually being…

We study the fluctuations of certain random matrix products $\Pi_N=M_N\cdots M_2M_1$ of $\mathrm{SL}(2,\mathbb{R})$, describing localisation properties of the one-dimensional Dirac equation with random mass. In the continuum limit, i.e.…

We prove that the local eigenvalue statistics in the bulk for complex random matrices with independent entries whose $r$-th absolute moment decays as $N^{-1-(r-2)\epsilon}$ for some $\epsilon>0$ are universal. This includes sparse matrices…

Consider the product of $m$ independent $n\times n$ random matrices from the spherical ensemble for $m\ge 1$. The spectral radius is defined as the maximum absolute value of the $n$ eigenvalues of the product matrix. When $m=1$, the…

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrt{\gamma})^2$…

In this paper we consider $N \times N $ matrices $D_{N}$ with i.i.d. entries all following an $a-$stable law divided by $N^{1/a}$. We prove that the least singular value of $D_{N}$, multiplied by $N$, tends to the same law as in the…

We define the row product of K matrices of size d by n as a matrix of size d^K by n, whose row are entry-wise products of rows of these matrices. This construction arises in certain computer science problems. We study the question, to which…

Let $A$ be an $n \times n$ random matrix with iid entries over a finite field of order $q$. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - \alpha$ for some fixed $0 < \alpha…

We treat the problem of testing independence between m continuous variables when m can be larger than the available sample size n. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank…

We consider products of independent square random non-Hermitian matrices. More precisely, let $n\geq 2$ and let $X_1,\ldots,X_n$ be independent $N\times N$ random matrices with independent centered entries with variance $N^{-1}$. It was…

We show that a necessary and sufficient condition for the sum of iid random vectors to converge (under appropriate shifting and scaling) to a multivariate Gaussian distribution is that the truncated second moment matrix is slowly varying at…