A Berry-Esseen type inequality for convex bodies with an unconditional basis
Bo'az Klartag
我们提供了一个尖锐收敛速率在中心极限定理的随机向量与无条件，对数凹密度。论证依赖于凸区域上的Neumann Laplacian分析和措施的最优运输理论。
We provide a sharp rate of convergence in the central limit theorem for random vectors with an unconditional, log-concave density. The argument relies on analysis of the Neumann laplacian on convex domains and on the theory of optimal transportation of measures.
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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 (1988) for standardized martingales. As a consequence, a Berry-Esseen bound is obtained.
Berry-Esseen bounds in local limit theorems
本文在多维局部极限定理中，基于Lyapunov系数和密度最大值发展了Berry-Esseen型界。
We obtain Berry-Esseen type estimates for "nonconventional" expressions of the form $ξ_N=\frac 1{\sqrt N}\sum_{n=1}^N(F(X(q_1(n)),...,X(q_\ell(n)))-\bar F)$
A Berry-Esseen bound is obtained for self-normalized martingales under the assumption of finite moments. The bound coincides with the classical Berry-Esseen bound for standardized martingales. An example is given to show the optimality of the bound. Applications to Student's statistic and autoregressive process are also discussed.
The asymptotic Berry-Esseen constant for intervals
Berry-Esseen不等式中区间概率的渐近常数为sqrt（2/pi）。
In this paper, we prove a Berry--Esseen bound with optimal order for self-normalized sums of local dependent random variables under some mild dependence conditions. The proof is based on Stein's method and a randomized concentration inequality. As applications, we obtain optimal Berry--Esseen bounds for $m$-dependent random variables and graph dependency.
In this paper, we establish optimal Berry--Esseen bounds for the generalized $U$-statistics. The proof is based on a new Berry--Esseen theorem for exchangeable pair approach by Stein's method under a general linearity condition setting. As applications, an optimal convergence rate of the normal approximation for subgraph counts in Erdös--Rényi graphs and graphon-random graph is obtained.
We obtain a Berry-Esseen type bound for the distribution of the maximum likelihood estimator of the drift parameter for fractional Ornstein-uhlenbeck type process driven by sub-fractional Brownian motion.
Berry-Esseen bounds in the entropic central limit theorem
对于非独立同分布的和，建立了全变差和到正规律的相对熵距离的Berry-Esseen型界.随机变量。
Berry-Esseen bound of modularity in network
针对给定网络的特定划分，研究证明了网络规模增大时模块度的 Berry-Esseen 界与强大数定律。
For an Ornstein-Uhlenbeck process driven by fractional Brownian motion with Hurst index $H\in [\frac12,\frac34]$, we show the Berry-Esséen bound of the least squares estimator of the drift parameter. We use an approach based on Malliavin calculus given by Kim and Park \cite{kim 3}.
Non-uniform Berry-Esseen bounds via Malliavin-Stein method
本文利用Malliavin-Stein方法建立非均匀Berry-Esseen界，并应用于多重Wiener-Itô积分与布朗运动指数泛函。
Under correlation-type conditions, we derive upper bounds of order $\frac{1}{\sqrt{n}}$ for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law.
We obtain a sharp estimate of the speed of convergence in the Boolean central limit theorem for measures of finite sixth moment. The main tool is a quantitative version of the Stieltjes-Perron inversion formula.
We establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a Poisson measure. Our results generalize previous findings by Nourdin and Peccati (2012, 2015) and Biermé, Bonami, Nourdin and Peccati (2013), involving random variables living on a Gaussian space. Applications are given to optimal Berry-Esseen bounds for edge counting in random geometric graphs.
We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cramér type large deviations for moderate $x$'s, and are of exponential decay rate as de la Peña's inequality when $x\rightarrow \infty$. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.
We derive explicit Berry-Esseen bounds in the total variation distance for the Breuer-Major central limit theorem, in the case of a subordinating function $φ$ satisfying minimal regularity assumptions. Our approach is based on the combination of the Malliavin-Stein approach for normal approximations with Gebelein's inequality, bounding the covariance of functionals of Gaussian fields in terms of maximal correlation coefficients.
We give estimates on the rate of convergence in the Boolean central limit theorem for the Lévy distance. In the case of measures with bounded support we obtain a sharp estimate by giving a qualitative description of this convergence.
Berry-Esseen bounds for estimators of entropy and diversity indices on countable alphabets
本文推导了可数字母表上多样性指数估计量的Berry-Esseen界，建立了广泛指数类插件估计量的非渐近收敛速率，包括Simpson指数和Renyi熵。针对Shannon熵的实用重要情况，给出了标准插件估计量以及两种常用偏置修正变体的显式Berry-Esseen界。
On rates of convergence and Berry-Esseen bounds for random sums of centered random variables with finite third moments
我们表明，如何经典的Berry-Esseen定理正常的近似可能被用来获得centerd，实值随机变量的随机和的收敛速度相对于某一类的概率距离，包括Kolmogorov和Wasserstein距离。这种技术适用于几个例子，包括拉普拉斯分布的几何和中心的随机变量与有限的三阶矩，其中一个具体的Berry-Esseen界的近似。这个界限简化为i.i.d.中的假定最优阶$\sqrt{p}$的界限。案子
An Inductive Proof of the Berry-Esseen Theorem for Character Ratios
Bolthausen使用了Stein方法的一种变体，对独立同分布随机变量的Berry-Esseen定理进行了归纳证明。我们修改这种技术证明了一个贝里-埃辛定理的随机表示的对称群的转置字符比。对于划分上的Jack测度也证明了类似的结果。
Two friendly proofs of the Berry-Esseen theorem
本文通过直观的傅里叶分析方法，给出了伯德-埃森定理的非渐近形式证明，适用于基础概率课程。
Berry-Esseen inequality for random walks conditioned to stay positive
本文针对条件为保持正的随机游走，推导Berry-Esseen型估计，证明了在增量零均值、有限方差下，其收敛于瑞利分布的速率为$n^{-1/2}$阶。
We give rates of convergence in the Central Limit Theorem for the coefficients and the spectral radius of the left random walk on GLd(R), assuming the existence of an exponential or polynomial moment.
We establish a Berry--Esseen bound for general multivariate nonlinear statistics by developing a new multivariate-type randomized concentration inequality. The bound is the best possible for many known statistics. As applications, Berry--Esseen bounds for M-estimators and averaged stochastic gradient descent algorithms are obtained.
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 distance between these standardized functionals and the standard normal distribution. Our approach relies on closed-form expressions for the action of some Malliavin type operators on avoidance functionals of Poisson processes. As a result we provide Berry-Esseen bounds in the CLT for the volume of the union of balls of a fixed radius around random Poisson cente…
Improved nonuniform Berry--Esseen-type bounds
基于非线性统计中关于非一致Berry-Esseen型界的研究，得到了独立随机变量和的新的非一致Berry-Esseen型界.证明是基于陈-邵浓度技术的框架内的斯坦方法。
Let $(Z_n)$ be a supercritical branching process in a random environment $ξ= (ξ_n)$. We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for $\log Z_n$ under the annealed law $\mathbb P$. We also improve some earlier results about the harmonic moments of the limit variable $W=lim_{n\to \infty} W_n$, where $W_n =Z_n/ \mathbb{E}_ξ Z_n$ is the normalized population size.
The Berry-Esseen bound in de Jong's CLT
本文针对归一化完全退化U-统计量，在四阶矩收敛于3和Lindeberg-Feller型条件下证明了de Jong中心极限定理的Berry-Esseen界，并达到与已有Wasserstein距离界相同的最优阶。
The Berry-Esseen Constant Conjecture is Eventually True
本文证明在样本量足够大时，Berry--Esseen 不等式中的 Esseen 常数是最优的，并给出通用阈值 $N=2\lceil\exp(10^{17})\rceil$。
We obtain Berry-Esseen-type bounds for the sum of random variables with a dependency graph and uniformly bounded moments of order $δ\in (2,\infty]$ using a Fourier transform approach. Our bounds improve the state-of-the-art in the regime where the degree of the dependency graph is large. As a Corollary of our results, we obtain a Central Limit Theorem for random variables with a sparse dependency graph that are uniformly bounded in $L^δ$ for some $δ\in(2,\infty]$.
Berry-Esseen bounds for non-linear functionals of infinite Rademacher sequences are derived by means of the Malliavin-Stein method. Moreover, multivariate extensions for vectors of Rademacher functionals are shown. The results establish a connection to small-ball probabilities and shed new light onto the relation between central limit theorems on the Rademacher chaos and norms of contraction operators. Applications concern infinite weighted 2-runs, a combinatorial central limit theorem and traces of Bernoulli random matrices.
On the Berry-Esseen bound for the Student statistic
新的Berry-Esseen型的界限，明确的常数因子，学生统计量的分布，并等价地，为独立零均值随机变量的自归一化和。这些界限与相应的现有结果进行了比较。
\noindent We study the asymptotic behavior of a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables. We prove a Berry-Esseen bound in a general setting and a large deviation result when the Laplace transform of the underlying distribution is not defined in a neighborhood of zero. Then we present several combinatorial applications. In particular, we prove a large deviation result for the model of hashing with linear probing.
A Berry-Esseen bound with applications to vertex degree counts in the Erdős-Rényi random graph
应用Stein的方法，一个归纳技术和大小偏置耦合产生一个正常的近似没有通常的限制，耦合是有界的Berry-Esseen定理。该定理被应用于计算给定度的Erdos-Renyi随机图的顶点数。
Non-uniform Berry-Esseen theorems for weakly dependent random variables
本文研究弱依赖序列的非均匀Berry-Esseen估计，涵盖均匀椭圆非齐马克诺夫链、随机时间变化的弱混沌动力系统、随机矩阵乘积及局部统计等类别。
Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given $3 < p \leq 4$ moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metr…
In this work, we provide a $(n/m)^{-1/2}$-rate finite sample Berry-Esseen bound for $m$-dependent high-dimensional random vectors over the class of hyper-rectangles. This bound imposes minimal assumptions on the random vectors such as nondegenerate covariances and finite third moments. The proof uses inductive relationships between anti-concentration inequalities and Berry--Esseen bounds, which are inspired by the telescoping method of Chen and Shao (2004) and the recursion method of Kuchibhotla and Rinaldo (2020). Performing a dual induction based on the relationships, we obtain tight Berry-E…
A Sub-Gaussian Berry-Esseen Theorem for the Hypergeometric Distribution
本文给出了超几何分布的参数弱收敛于正态极限的一个充要条件。仅在这个充要条件下，我们建立了超几何分布的Berry-Esseen定理.我们进一步推导出一个非均匀的Berry-Esseen界，其中超几何分布和正态分布函数之间的差异的尾部以亚高斯速率衰减。
Non-uniform Berry--Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and Jack measures
本文借助 Stein 方法与集中不等式建立可交换对的非一致 Berry–Esseen 界，改进已有定理，并将其应用于平均场经典 $N$-向量模型和 Jack 测度。
Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices and consider the matrix product $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and moderate deviation expansions of Cramér type, for the matrix norm $\| G_n \|$ of $G_n$, for its $(i,j)$-th entry $G_n^{i,j}$, and the and for its spectral radius $ρ(G_n)$.
Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random variable. Given sufficiently moments, when do we have a rate of convergence of $n^{-1/2}$ in the uniform metric, in other words, when do we have the optimal Berry-Esseen bound? We study this question in a quite general framework and find the (almost) sharp dependence conditions. The result applies to many different processes and dynamical systems. As specific, prominent examples, we study functions of the doubling map 2x mod 1 and the left random walk on the general linear group.
As an extension of a central limit theorem established by Svante Janson, we prove a Berry-Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.
We study the inhomogeneous Curie-Weiss model with external field, where the inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry-Esseen rate of convergence for this limit theorem using Stein's method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables.
Berry-Esseen bound for the Brownian motions on hyperbolic spaces
本文研究了双曲空间上布朗运动径向部分的高斯波动，基于Millson公式和分部积分公式得到了均匀收敛速率，并证明在维数为2或一般奇数时该结果是精确的。
Variations on the Berry-Esseen theorem
我们分析了有限四阶矩的n个独立同分布随机变量的线性组合的高斯近似的质量。事实证明，存在普遍的、简单的线性组合，它们比变量之和表现得更好。我们还研究了随机变量是独立的，但它们不一定是同分布的情况。
Hwang's quasi-power theorem asserts that a sequence of random variables whose moment generating functions are approximately given by powers of some analytic function is asymptotically normally distributed. This theorem is generalised to higher dimensional random variables. To obtain this result, a higher dimensional analogue of the Berry-Esseen inequality is proved, generalising a two-dimensional version by Sadikova.
More on the nonuniform Berry--Esseen bound
最近发表在arxiv上的论文arxiv：1301.2828中提出了某些平滑不等式，以缩小非均匀Berry-Esseen界中常数因子的最佳正确建立的上限和下界之间的巨大差距。为了说明这些不等式的可能用途，给出了Nagaev经典非一致界的一个快速证明。在这里，我们描述了另一个，显然更有效的一类光滑不等式，并给出了一个更快，更简单的证明Nagaev的结果。
Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing
Berry Esseen型边界的正常，基于零和大小偏置耦合，使用斯坦的方法。零偏界的组合中心极限定理的随机置换具有均匀分布或一个是恒定的置换具有相同的周期类型，并没有不动点的应用程序进行说明。大小偏置界限被应用于固定的相对有序的子序列（如上升序列）在随机排列中的出现，以及有限图上的模式、极值和子图的出现。
A new Berry-Esseen bound for non-linear functionals of non-symmetric and non-homogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin-Stein method and an analysis of the discrete Ornstein-Uhlenbeck semigroup. The result is applied to sub-graph counts and to the number of vertices having a prescribed degree in the Erdős-Renyi random graph. A further application deals with a percolation problem on trees.
It is shown that the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if $1\le n\le 500000$, where $n$ is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009--2011, an upper bound is obtained for the absolute constant in the Berry--Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.0…
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. Probab. 1982) and Grama and Haeusler (Stochastic Process. Appl. 2000), is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.
Berry-Esseen for Free Random Variables
本文证明了Berry-Esseen不等式的一个类似形式，即关于有界概率测度的自由可加卷积的收敛速度.得到的收敛速率为n^{-1/2}阶，与经典情形相同。一个二项测度的例子表明，如果不对卷积测度施加进一步的限制，这种估计就不能得到改进。
We prove Berry-Esseen type rates of convergence for central limit theorems (CLTs) of regenerative processes which generalize previous results of Bolthausen under weaker moment assumptions. We then show how this general result can be applied to obtain rates of convergence for (1) CLTs for additive functionals of positive recurrent Markov chains under certain conditions on the strong mixing coefficients, and (2) annealed CLTs for certain ballistic random walks in random environments.
We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on chaos expansions methods. Our results apply to $U$-statistics satisfying the weak assumption of decomposability in the Hoeffding sense, and yield Kolmogorov distance bounds instead of the Wasserstein bounds previously derived in the special case of degenerate $U$-statistics. Linear and quadratic functionals of arbitrary sequences of independent random variables are included as particular cases, with new fourth moment bounds, and applications are given to Hoeffding decompositions, weigh…
In the context of bounding probability of small deviation, there are limited general tools. However, such bounds have been widely applied in graph theory and inventory management. We introduce a common approach to substantially sharpen such inequality bounds by combining the semidefinite optimization approach of moments problem and the Berry-Esseen theorem. As an application, we improve the lower bound of Feige's conjecture from 0.14 to 0.1798.
A new Berry-Esseen-type estimate in the free central limit theorem
本文利用从属方法，基于第四Lyapunov分数在自由中心极限定理中提出了一种新的Berry-Esseen型估计，在相同分布下几乎达到了最优速率。
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.
Let $A_n= \varepsilon_n \cdots \varepsilon_1$, where $(\varepsilon_n)_{n \geq 1}$ is a sequence of independent random matrices taking values in $ GL_d(\mathbb R)$, $d \geq 2$, with common distribution $μ$. In this paper, under standard assumptions on $μ$ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for $\log ( \Vert A_n \Vert)$ when $μ$ has a polynomial moment. More precisely, we get the rate $((\log n) / n)^{q/2-1}$ when $μ$ has a moment of order $q \in ]2,3]$ and the rate $1/ \sqrt{n} $ when $μ$ has a moment of order $4$, which significantly improves earlier r…
We consider a branching random walk on $d$-dimensional real space with immigration in a time-dependent random environment. Let $Z_n(\mathbf t)$ be the so-called partition function of the process, namely, the moment generating function of the counting measure describing the dispersion of individuals at time $n$. For $\mathbf t$ fixed, the logarithm $\log Z_n(\mathbf t)$ satisfies a central limit theorem. By studying the logarithmic moments of the intrinsic submartingale of the system and its convergence rates, we establish the uniform and non-uniform Berry-Esseen bounds corresponding to the cen…
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