Related papers: Berry-Esseen bounds in the entropic central limit …

Renz (Ann. Probab. 1996) has established a rate of convergence $1/\sqrt{n}$ in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen (Ann.…

We give a nonuniform Berry-Esseen bound for self-normalized martingales, which bridges the gap between the result of Haeusler (1988) and Fan and Shao (2018). The bound coincides with the nonuniform Berry-Esseen bound of Haeusler and Joos…

A central limit theorem for arrays of symmetric row-wise exchangeable random variables is presented. The result is valid for finite and infinite extendable and non-extendable sequences. Unlike most reported versions of the central limit…

Due to the effort of a number of authors, the value c_u of the absolute constant factor in the uniform Berry--Esseen (BE) bound for sums of independent random variables has been gradually reduced to 0.4748 in the iid case and 0.5600 in the…

Fix a container polygon $P$ in the plane and consider the convex hull $P_n$ of $n\geq 3$ independent and uniformly distributed in $P$ random points. In the focus of this paper is the vertex number of the random polygon $P_n$. The precise…

We study rates of convergence in central limit theorems for the partial sum of squares of general Gaussian sequences, using tools from analysis on Wiener space. No assumption of stationarity, asymptotically or otherwise, is made. The main…

A Berry--Ess\'{e}en theorem for linear combinations of iterates of an inner function is obtained. Our proof, which is based an elementary transfer argument and classical results in martingale theory, also leads to a simple proof of Nicolau…

This paper deals with the union set of a stationary Poisson process of cylinders in $\mathbb{R}^n$ having an $(n-m)$-dimensional base and an $m$-dimensional direction space, where $m\in\{0,1,\ldots,n-1\}$ and $n\geq 2$. The concept…

We prove that, in Young towers with sufficiently small tails, the speed in the central limit theorem is O(1/\sqrt{n}), and the local limit theorem holds. This implies the same results for many non uniformly expanding dynamical systems,…

We consider functionals which are weighted averages of the avoidance function of a Poisson process. Using the approach to Stein's method based on Malliavin calculus for Poisson functionals we provide explicit bounds for the Wasserstein…

For any integer $m<n$, where $m$ can depend on $n$, we study the rate of convergence of $\frac{1}{\sqrt{m}}\mathrm{Tr} \mathbf{U}^m$ to its limiting Gaussian as $n\to\infty$ for orthogonal, unitary and symplectic Haar distributed random…

We prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is…

We study higher order expansions both in the Berry-Ess\'een estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical…

In this note, we provide a Berry--Esseen bounds for rectangles in high-dimensions when the random vectors have non-singular covariance matrices. Under this assumption of non-singularity, we prove an $n^{-1/2}$ scaling for the Berry--Esseen…

A quantitative central limit theorem for the simple symmetric exclusion process (SSEP) on a $d$-dimensional discrete torus is proven. The argument is based on a comparison of the generators of the density fluctuation field of the SSEP and…

Applying an inductive technique for Stein and zero bias couplings yields Berry-Esseen theorems for normal approximation for two new examples. The conditions of the main results do not require that the couplings be bounded. Our two…

The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic dynamics developed by the first author \textit{et al}. to establish quenched versions of the large deviation…

Homogeneous normalized random measures with independent increments (hNRMIs) represent a broad class of Bayesian nonparametric priors and thus are widely used. In this paper, we obtain the strong law of large numbers, the central limit…

Let $B$ be a bifractional Brownian motion with parameters $H\in (0, 1)$ and $K\in(0,1]$. For any $n\geq1$, set $Z_n =\sum_{i=0}^{n-1}\big[n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\E((B_{i+1}-B_{i})^2)\big]$. We use the Malliavin calculus and the…

We give a distribution-dependent concentration inequality for functions of independent variables. The result extends Bernstein's inequality from sums to more general functions, whose variation in any argument does not depend too much on the…