Related papers: Central Limit Theorems and Bootstrap in High Dimen…

In this work we study the rate of convergence in the central limit theorem for the Euclidean norm of random orthogonal projections of vectors chosen at random from an $\ell_p^n$-ball which has been obtained in [Alonso-Guti\'errez, Prochno,…

When the underlying random variables are Gaussian, the classical Central Limit Theorem (CLT) is trivial, but the functional CLT is not. The objective of the paper is to investigate the functional CLT for stationary Gaussian processes in the…

We consider the Hopfield model with $n$ neurons and an increasing number $p=p(n)$ of randomly chosen patterns and use Stein's method to obtain rates of convergence for the central limit theorem of overlap parameters, which holds for every…

Consider the sample covariance matrix $$\Sigma^{1/2}XX^T\Sigma^{1/2}$$ where $X$ is an $M\times N$ random matrix with independent entries and $\Sigma$ is an $M\times M$ diagonal matrix. It is known that if $\Sigma$ is deterministic, then…

We consider the classification problem of a high-dimensional mixture of two Gaussians with general covariance matrices. Using the replica method from statistical physics, we investigate the asymptotic behavior of a general class of…

Let $\nu\in M^1([0,\infty[)$ be a fixed probability measure. For each dimension $p\in\b N$, let $(X_n^p)_{n\ge1}$ be i.i.d. $\b R^p$-valued radial random variables with radial distribution $\nu$. We derive two central limit theorems for $…

In this paper we show that the limiting distribution of the real and the imaginary part of the double Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance…

In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $\mu$ and $\nu$ such that $\nu$ is log-concave with respect to $\mu$.…

This paper considers a distributionally robust chance constraint model with a general ambiguity set. We show that a sample based approximation of this model converges under suitable sufficient conditions. We also show that upper and lower…

In high-dimensional regression, we attempt to estimate a parameter vector $\beta_0\in\mathbb{R}^p$ from $n\lesssim p$ observations $\{(y_i,x_i)\}_{i\leq n}$ where $x_i\in\mathbb{R}^p$ is a vector of predictors and $y_i$ is a response…

This paper gives a theoretical analysis of high dimensional linear discrimination of Gaussian data. We study the excess risk of linear discriminant rules. We emphasis on the poor performances of standard procedures in the case when…

Choose $n$ random, independent points in $\R^d$ according to the standard normal distribution. Their convex hull $K_n$ is the {\sl Gaussian random polytope}. We prove that the volume and the number of faces of $K_n$ satisfy the central…

We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $\alpha$-stable laws and…

In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $\ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large…

In the context of principal components analysis (PCA), the bootstrap is commonly applied to solve a variety of inference problems, such as constructing confidence intervals for the eigenvalues of the population covariance matrix $\Sigma$.…

In this paper we obtain the central limit theorem for triangular arrays of non-homogeneous Markov chains under a condition imposed to the maximal coefficient of correlation. The proofs are based on martingale techniques and a sharp lower…

We give a general local central limit theorem for the sum of two independent random variables, one of which satisfies a central limit theorem while the other satisfies a local central limit theorem with the same order variance. We apply…

This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to $C^3$, the class of functions with continuous third order derivatives. We…

A random geometric digraph $G_n$ is constructed by taking $\{X_1,X_2,... X_n\}$ in $\mathbb{R}^2$ independently at random with a common bounded density function. Each vertex $X_i$ is assigned at random a sector $S_i$ of central angle…

The martingale posterior framework is a generalization of Bayesian inference where one elicits a sequence of one-step ahead predictive densities instead of the likelihood and prior. Posterior sampling then involves the imputation of unseen…