Related papers: Functional CLT for sample covariance matrices

In this paper we provide an asymptotic theory for the symmetric version of the Kullback--Leibler (KL) divergence. We define a estimator for this divergence and study its asymptotic properties. In particular, we prove Law of Large Numbers…

In this paper, our objective is to present a constraining principle governing the spectral properties of the sample covariance matrix. This principle exhibits harmonious behavior across diverse limiting frameworks, eliminating the need for…

We establish effective convergence rates in the Doeblin-Lenstra law, describing the limiting distribution of approximation coefficients arising from continued fraction convergents of a typical real number. More generally, we prove…

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise…

Using Stein's method, we prove an abstract result that yields multivariate central limit theorems with a rate of convergence for time-dependent dynamical systems. As examples we study a model of expanding circle maps and a quasistatic…

The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$ are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the…

High-dimensional sample correlation matrices are a crucial class of random matrices in multivariate statistical analysis. The central limit theorem (CLT) provides a theoretical foundation for statistical inference. In this paper, assuming…

For large dimensional non-Hermitian random matrices $X$ with real or complex independent, identically distributed, centered entries, we consider the fluctuations of $f(X)$ as a matrix where $f$ is an analytic function around the spectrum of…

This article provides a central limit theorem for a consistent estimator of population eigenvalues with large multiplicities based on sample covariance matrices. The focus is on limited sample size situations, whereby the number of…

Let $\mathbf{B}_n=\mathbf {S}_n(\mathbf {S}_n+\alpha_n\mathbf {T}_N)^{-1}$, where $\mathbf {S}_n$ and $\mathbf {T}_N$ are two independent sample covariance matrices with dimension $p$ and sample sizes $n$ and $N$, respectively. This is the…

Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function…

We derive a Central Limit Theorem (CLT) for $\log \left\vert\det \left( W_{N}-E_{N}\right)\right\vert,$ where $W_{N}$ is a Wigner matrix, and $E_{N}$ is local to the edge of the semi-circle law. Precisely, $E_N=2+N^{-2/3}\sigma_N$ with…

The approximation of integral type functionals is studied for discrete observations of a continuous It\^o semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions…

We prove a central limit theorem (CLT) for the Frechet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Frechet mean is unique. Previous general CLT results in this…

We derive asymptotic formulas for central extended binomial coefficients, which are generalizations of binomial coefficients. To do so, we relate the exact distribution of the sum of independent discrete uniform random variables to the…

A nonparametric model using a sequence of Bernstein polynomials is constructed to approximate arbitrary isotropic covariance functions valid in $\mathbb{R}^\infty$ and related approximation properties are investigated using the popular…

We consider the asymptotics of various estimators based on a large sample of branching trees from a critical multi-type Galton-Watson process, as the sample size increases to infinity. The asymptotics of additive functions of trees, such as…

In this paper, we propose corrections to the likelihood ratio test and John's test for sphericity in large-dimensions. New formulas for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with…

We estimate linear functionals in the classical deconvolution problem by kernel estimators. We obtain a uniform central limit theorem with $\sqrt{n}$-rate on the assumption that the smoothness of the functionals is larger than the…

We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n \{f(X_k)-\E[f(X_k)]\}$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It…