The multivariate Berry-Esseen theorem with an absolute constant

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1. Very close Another approach to the problem Cited by this paper Used in depth by this paper Not relevant

   A multivariate Berry–Esseen theorem with explicit constants

   M. Raič · 2018 · Bernoulli · cited 113 times · Not summarized yet

   A multivariate Berry–Esseen theorem with explicit constants bounds normal approximation over convex sets for independent vectors, using Stein’s method and explicit constants.

   Abstract

   We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein’s method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.

2. Very close Another approach to the problem Not relevant

   Improved Berry-Esseen bounds for multivariate nonlinear statistics in convex distance

   Zhi-Jun Cai, Jing-Cai Yang, Zhuo-Song Zhang · 2026 · cited 1 time · Not summarized yet

   Improved Berry-Esseen bounds for multivariate nonlinear statistics treats Gaussian approximation over convex sets, including independent-sum contributions and a dependent remainder.

   Abstract

   In this paper, we establish two nonasymptotic Berry--Esseen bounds over convex sets for the Gaussian approximation of multivariate nonlinear statistics. The statistics of interest can be written as a sum of independent centered random vectors plus a remainder that may depend on all observations. The first bound retains the classical factor $d^{1/4}$ in the contribution of the independent sum, where $d$ is the dimension, while controlling the remainder through its size and its sensitivity to replacing a single observation. The second bound expresses the contribution of the independent sum in terms of fourth moments and can allow the dimension to grow faster with the sample size. For sums of i…

3. Close Broader context Not relevant

   The Berry--Esseen estimate in the Boolean central limit theorem

   M. Maejima, Noriyoshi Sakuma · 2026 · Not summarized yet

   The Berry–Esseen estimate in the Boolean central limit theorem studies convergence rates for Boolean convolution powers to a Bernoulli law, rather than Gaussian approximation of vector sums.

   Abstract

   We estimate the L\'evy distance to the symmetric Bernoulli distribution for normalized Boolean convolution powers of a probability measure with mean zero and variance one. If the absolute moment of order $r$ is finite, we obtain the upper bounds $o(n^{-(r-2)/6})$ for $2<r<5$ and $O(n^{-1/2})$ for $r\ge5$. The proof relies on an inequality that controls the L\'evy distance to the Bernoulli law by the values of the reciprocal Cauchy transform at two points. For each $r\ge2$, we also construct a single fixed distribution whose absolute moments are finite exactly up to order $r$, and show that the exponent $\min\{(r-2)/6,1/2\}$ cannot be improved. Under only a finite second moment assumption, th…

4. Close Broader context Not relevant

   Gaussian Lower Bounds for Distinct Subset Sums in Higher Dimensions

   Zi-Jie Gu · 2026 · Not summarized yet

   Gaussian Lower Bounds for Distinct Subset Sums uses Gaussian-kernel comparisons in a higher-dimensional combinatorial problem, not normal approximation for independent sums.

   Abstract

   For sequences of $n$ nonnegative integer vectors in $\mathbb Z^k$ with distinct subset sums, we prove, for fixed $k$, the uniform lower bound $M\ge (C_k-o(1))2^{n/k}/\sqrt n$ for the largest coordinate, where \[ C_k=\sqrt{\frac{2k}{\pi}}\, \frac{(k+1)^{(k-1)/(2k)}}{\sqrt{\lfloor (k+1)^2/4\rfloor}}. \] The proof compares a Gaussian-kernel expectation with a sum over a translated lattice, uniformly over the vector sequence. We optimise the resulting quadratic-form constant using a vertex second-moment certificate. We show that $C_k$ also exceeds the variance constant obtained by optimising an ellipsoidal geometry.

5. Close Broader context Not relevant

   $L^p$ maximal inequalities for Gaussian subordination with applications to Breuer--Major--Donsker principles

   D. Milesis, Guang-Qu Zheng · 2026 · cited 1 time · Not summarized yet

   L^p maximal inequalities for Gaussian subordination bounds partial sums of dependent Gaussian-subordinated variables, rather than Berry–Esseen approximation for independent vectors.

   Abstract

   In this paper, we establish an $L^2$ maximal inequality for partial sums of Gaussian-subordinated random variables. More precisely, let $f:\mathbb{R}\to\mathbb{R}$ belong to $L^2(\gamma)$ and have Hermite rank at least $d$, where $\gamma$ denotes the standard Gaussian measure. If the covariance matrix of the underlying Gaussian family, with standard normal marginals, satisfies a uniform $d$th-power row bound $K$, then the $L^2$ norm of the maximal partial sum is bounded by $C_{d,K}\sqrt n\,\|f\|_{L^2(\gamma)}$. The constant $C_{d,K}$ is independent of $(f,n)$ and depends on the covariance structure only through $K$. As a consequence, our $L^2$ maximal inequality controls Hermite tails unifor…

6. Close* Same question Cited by this paper Not relevant

   Dependence of the Berry-Esseen estimate on the dimension

   V. Bentkus · 1986 · Lithuanian Mathematical Journal · cited 38 times · Not summarized yet

   Dependence of the Berry-Esseen estimate on the dimension addresses how dimension affects multivariate Berry–Esseen bounds.

7. Close Broader context Not relevant

   The sharp exponent for the Banach--Mazur distance to the cube

   O. Friedland · 2026 · Not summarized yet

   The sharp exponent for the Banach–Mazur distance to the cube uses Gaussian polytopes and projection arguments to study distances between normed spaces.

   Abstract

   Let $R_\infty(n)$ be the maximal Banach--Mazur distance from an $n$-dimensional real normed space to $\ell_\infty^n$. We prove that $R_\infty(n)\ge cn^{2/3}(\log(2n))^{-10}$. Together with the upper bound of Friedland, Naor, and Youssef, this gives $R_\infty(n) = n^{2/3+o(1)}$. The examples are polars of symmetric Gaussian polytopes with $2n$ generators, and the lower bound holds with exponentially high probability. A joint minimization over coefficient representations and positive weights yields an exact second-variation identity and sparse coefficients. A Gaussian projection argument controls all low-rank completions of the recorded matrix. An adaptive choice of exceptional generators then…

8. Close Broader context Not relevant

   Approximate independence of typical marginals of high-dimensional probability measures

   Manuel Arnese, Daniel Lacker · 2026 · Not summarized yet

   Approximate independence of typical marginals studies product-like coordinate marginals of high-dimensional measures, rather than Gaussian approximation for sums.

   Abstract

   For many probability measures on $\mathbb{R}^n$, and for $k$ sufficiently smaller than $n$, most $k$-dimensional coordinate marginals are close to product measures. We prove this in two regimes, which notably involve no exchangeability assumptions as in prior theorems of de Finetti type. First, for log-concave measures, the quantitative bound depends on the measure only through the operator norm of its correlation matrix, and $k$ may grow polynomially with $n$. A key ingredient of the proof is a new strengthening of the subadditivity of relative entropy for strongly log-concave measures, which as a byproduct yields sharp bounds in the strongly log-concave case. Second, for a broad class of m…

Prior work it builds on

1. Related Another approach to the problem Not relevant

   Multivariate Normal Approximation by Stein's Method: The Concentration Inequality Approach

   Louis H. Y. Chen, Xiao Fang · 2011 · arXiv: Probability · preprint, not peer reviewed · cited 40 times · Not summarized yet

   Multivariate Normal Approximation by Stein's Method bounds normal approximation for independent vector sums using a concentration-inequality approach and a dimension-dependent error.

   Abstract

   The concentration inequality approach for normal approxima- tion by Stein's method is generalized to the multivariate setting. We use this approach to prove a non-smooth function distance for multivariate normal approximation for standardized sums of k-dimensional indepen- dent random vectors W = Pn=1 Xi with an error bound of order k 1/2 where = P n=1 E|Xi| 3 . For sums of locally dependent (unbounded) ran- dom vectors, we obtain a fourth moment bound which is typically of order Ok(1/ p n), as well as a third moment bound which is typically of order

2. Related* Groundwork for this paper Not relevant

   An estimate of the remainder term in the multidimensional central limit theorem

   S. Nagaev · 1976 · cited 40 times · Not summarized yet

   An estimate of the remainder term in the multidimensional central limit theorem is earlier work on error bounds for multidimensional normal approximation.

3. Related* Groundwork for this paper Not relevant

   The accuracy of the Gaussian approximation to the sum of independent variates

   Andrew C. Berry · 1941 · Transactions of the American Mathematical Society · cited 846 times · Not summarized yet

   The accuracy of the Gaussian approximation to the sum of independent variates is earlier work on Gaussian approximation for sums of independent variables.

4. Related* Groundwork for this paper Not relevant

   On the Rate of Convergence in the Multivariate CLT

   F. Götze · 1991 · Annals of Probability · cited 335 times · Not summarized yet

   On the Rate of Convergence in the Multivariate CLT is earlier work on convergence rates in the multivariate central limit theorem.

5. Related* Groundwork for this paper Not relevant

   A Lyapunov type bound in $R^d$@@@A Lyapunov-type bound in $R^d$

   V. Bentkus, V. Bentkus · 2004 · cited 42 times · Not summarized yet

   A Lyapunov-type bound in R^d is earlier work on dimension-specific bounds for multivariate normal approximation.