Related papers: Logarithmic law of large random correlation matric…
For a given $p\times n$ data matrix $\textbf{X}_n$ with i.i.d. centered entries and a population covariance matrix $\bf{\Sigma}$, the corresponding sample precision matrix $\hat{\bf\Sigma}^{-1}$ is defined as the inverse of the sample…
Consider a $N\times n$ matrix $\Sigma_n=\frac{1}{\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear…
Let X_n=(x_{ij}) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let R_n=(\rho_{ij}) be the p\times p sample correlation matrix of X_n; that is, the entry…
If the log likelihood is approximately quadratic with constant Hessian, then the maximum likelihood estimator (MLE) is approximately normally distributed. No other assumptions are required. We do not need independent and identically…
We prove two theorems related to the Central Limit Theorem (CLT) for Martin-L\"of Random (MLR) sequences. Martin-L\"of randomness attempts to capture what it means for a sequence of bits to be "truly random". By contrast, CLTs do not make…
A key feature of a sequential study is that the actual sample size is a random variable that typically depends on the outcomes collected. While hypothesis testing theory for sequential designs is well established, parameter and precision…
Consider two high-dimensional random vectors $\widetilde{\mathbf x}\in\mathbb R^p$ and $\widetilde{\mathbf y}\in\mathbb R^q$ with finite rank correlations. More precisely, suppose that $\widetilde{\mathbf x}=\mathbf x+A\mathbf z$ and…
We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables $\xi_j$, perturbed by predictable multiplicative factors $\lambda_j$ with values in intervals $[\underline\lambda_j,\overline\lambda_j]$. It…
Consider $k$ independent random samples from $p$-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of $k$ covariance…
It is well known that central order statistics exhibit a central limit behavior and converge to a Gaussian distribution as the sample size grows. This paper strengthens this known result by establishing an entropic version of the CLT that…
It is known that the fluctuations of suitable linear statistics of Haar distributed elements of the compact classical groups satisfy a central limit theorem. We show that if the corresponding test functions are sufficiently smooth, a rate…
Symmetry is a cornerstone of much of mathematics, and many probability distributions possess symmetries characterized by their invariance to a collection of group actions. Thus, many mathematical and statistical methods rely on such…
Sample correlation matrices are employed ubiquitously in statistics. However, quite surprisingly, little is known about their asymptotic spectral properties for high-dimensional data, particularly beyond the case of "null models" for which…
This paper deals with subspace estimation in the small sample size regime, where the number of samples is comparable in magnitude with the observation dimension. The traditional estimators, mostly based on the sample correlation matrix, are…
We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble…
We consider the ensemble of N-dimensional random symmetric matrices A that have, in average, p non-zero elements per row. We study the asymptotic behavior of the norm of A in the limit of infinitely increasing N and p. We prove that the…
Let $G=G(n,p_n)$ be a homogeneous Erd\"os-R\'enyi graph, and $A$ its adjacency matrix with eigenvalues $\lambda_1(A) \geq \lambda_2(A) \geq ... \geq \lambda_n(A).$ Local laws have been used to show that $lambda_2(A)$ can exhibit…
We analyze the asymptotic fluctuations of linear eigenvalue statistics of random centrosymmetric matrices with i.i.d. entries. We prove that for a complex analytic test function, the centered and normalized linear eigenvalue statistics of…
In this paper, we consider the separable covariance model, which plays an important role in wireless communications and spatio-temporal statistics and describes a process where the time correlation does not depend on the spatial location…
Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i<j\leq…