Esseen type bounds of the remainder in a combinatorial CLT

Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices

We provide bounds of Berry-Esseen type for fundamental limit theorems in operator-valued free probability theory such as the operator-valued free Central Limit Theorem and the asymptotic behaviour of distributions of operator-valued matrices. Our estimates are on the level of operator-valued Cauchy transforms and the Lévy distance. We address the single-variable as well as the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the Lévy…

Berry-Esseen bounds for weighted averages of Poisson avoidance functionals

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…

Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes

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…

随机比值中心极限定理中的新 Kolmogorov 界及其应用

New Kolmogorov bounds in the CLT for random ratios and applications

本文针对 Gaussian 泛函比值建立基于三、四阶累积量的显式 Kolmogorov Berry–Esseen 界，并将其用于 OU 过程漂移参数估计量离散化分布的收敛速率分析，所得上界严格优于文献结果。

Monte Carlo estimation of the solution of fractional partial differential equations

The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem(CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.

乘积空间上正态逼近的新界及其在单色边、随机和与无穷 de Jong CLT 中的应用

New bounds for normal approximation on product spaces with applications to monochromatic edges, random sums and an infinite de Jong CLT

本文扩展了乘积概率空间上的 Malliavin 理论，结合 Stein 方法提出了三种抽象的 Berry-Esseen 界和 Wasserstein 界，并应用于单色边数量、随机和及无穷 de Jong CLT。

Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert space

Let $H$ be a real separable Hilbert space and $(a_k)_{k\in\mathbb{Z}}$ a sequence of bounded linear operators from $H$ to $H$. We consider the linear process $X$ defined for any $k$ in $\mathbb{Z}$ by $X_k=\sum_{j\in\mathbb{Z}}a_j(\varepsilon_{k-j})$ where $(\varepsilon_k)_{k\in\mathbb{Z}}$ is a sequence of i.i.d. centered $H$-valued random variables. We investigate the rate of convergence in the CLT for $X$ and in particular we obtain the usual Berry-Esseen's bound provided that $\sum_{j\in\mathbb{Z}}\vert j\vert\|a_j\|_{\mathcal{L}(H)}<+\infty$ and $\varepsilon_0$ belongs to $L_H^{\infty}$.

高维自归一化和的Berry-Esseen界

The Berry-Esseen Bound for High-dimensional Self-normalized Sums

该研究针对高维自归一化统计量逐坐标最大值的高斯近似问题，在弱矩假设下推导出显式Berry-Esseen界，填补了高维CLT领域的相关研究空白。

高温下β系综与可积系统的中心极限定理：一种转移算子方法

CLT for $β$ ensembles at high-temperature, and for integrable systems: a transfer operator approach

该研究采用转移算子方法，证明了若干可积模型与高温多项式势β系综的多项式中心极限定理，关联了两类系统Lax矩阵矩的统计量，证明可积系统局部函数空间相关指数衰减，并建立了Berry–Esseen型界。

通过有效RPF速率得到的非一致部分扩张随机动力系统的CLT显式条件及相关结果

Explicit conditions for the CLT and related results for non-uniformly partially expanding random dynamical systems via effective RPF rates

该研究为非一致部分扩张的非独立同分布随机变换与随机Gibbs测度构成的系统提供了首类CLT及相关渐近结果的显式充分条件，证明了对应随机Birkhoff和满足多项概率极限定理。

高维 m-依赖数据的双重归纳 CLT

Dual Induction CLT for High-dimensional m-dependent Data

本文研究高维 m-依赖随机向量和的 CLT 逼近，提出连接反集中与 Berry–Esseen 界的双重归纳方法，在弱矩条件下得到尖锐且近最优的样本复杂度。

Optimal rates of convergence for quenched central limit theorem rates for hitting times of one-dimensional random walks in random environments

We consider the rates of convergence of the quenched central limit theorem for hitting times of one-dimensional random walks in a random environment. Previous results had identified polynomial upper bounds for the rates of decay which are sometimes slower than $n^{-1/2}$ (the optimal rate in the classical Berry-Esseen estimates). Here we prove that the previous upper bounds are in fact the best possible polynomial rates for the quenched CLT.

凸距离下多变量非线性统计量的改进Berry-Esseen界

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

本文针对多变量非线性统计量在凸距离下建立两个非渐近Berry-Esseen界，改进维数增长条件，并应用于随机逼近、时间差分学习和U-统计量，给出显式高斯近似误差及收敛充分条件。

扩张映射序列的中心极限定理与几乎必然不变原理的收敛速率

Rates of convergence in CLT and ASIP for sequences of expanding maps

本文针对扩张映射序列驱动的部分和，建立了带收敛速率的Berry-Esseen定理与几乎必然不变原理，并将结果推广到Axiom A映射小邻域及Hölder连续函数情形。

On the integrated squared error of the linear wavelet density estimator

Linear wavelet density estimators are wavelet projections of the empirical measure based on independent, identically distributed observations. We study here the law of the iterated logarithm (LIL) and a Berry-Esseen type theorem. These results are proved under different assumptions on the density $f$ that are different from those needed for similar results in the case of convolution kernels (KDE): whereas the smoothness requirements are much less stringent than for the KDE, Riemann integrability assumptions are needed in order to compute the asymptotic variance, which gives the scaling constan…

Quantitative Estimates for Operator-Valued and Infinitesimal Boolean and Monotone Limit Theorems

We provide Berry-Esseen bounds for sums of operator-valued Boolean and monotone independent variables, in terms of the first moments of the summands. Our bounds are on the level of Cauchy transforms as well as the Lévy distance. As applications, we obtain quantitative bounds for the corresponding CLTs, provide a quantitative "fourth moment theorem" for monotone independent random variables including the operator-valued case, and generalize the results by Hao and Popa on matrices with Boolean entries. Our approach relies on a Lindeberg method that we develop for sums of Boolean/monotone indepen…

Ewens-Pitman模型的大数定律与中心极限定理

Laws of large numbers and central limit theorem for Ewens-Pitman model

本文研究Ewens-Pitman模型中块数$K_n$在参数$\theta$线性依赖于$n$时的渐近行为，建立大数定律和中心极限定理，表明$K_n$以$n$为尺度，并针对不同$\alpha$给出证明方法。

低信噪比下高维贝叶斯线性回归中的中心极限定理（CLT）

CLT in high-dimensional Bayesian linear regression with low SNR

该研究在低信噪比的高维贝叶斯线性回归非收缩机制下，证明线性统计量的极限分布为围绕后验平均场近似的高斯分布，无需先验稀疏假设，还构建了误设先验下的可信区间并计算其覆盖概率。

随机张量的反集中与Berry--Esseen界

Anticoncentration and Berry--Esseen bounds for random tensors

本文研究了随机张量线性函数的反集中与Berry-Esseen界，提出高维张量组合CLT作为关键技术，并在任意维度推广了Bolthausen及Barbour/Chen的经典结果。

中心极限定理与高维自归一化和的近似Berry-Esseen速率

Central Limit Theorem and Near classical Berry-Esseen rate for self normalized sums in high dimensions

本文研究高维自归一化和的中心极限定理及Berry-Esseen速率，证明多项式矩条件下的最优截断速率和最优Berry-Esseen速率。

Explicit rates of approximation in the CLT for quadratic forms

Let $X,X_1,X_2,\ldots$ be i.i.d. ${\mathbb{R}}^d$-valued real random vectors. Assume that ${\mathbf{E}X=0}$, $\operatorname {cov}X=\mathbb{C}$, $\mathbf{E}\Vert X\Vert^2=σ^2$ and that $X$ is not concentrated in a proper subspace of $\mathbb{R}^d$. Let $G$ be a mean zero Gaussian random vector with the same covariance operator as that of $X$. We study the distributions of nondegenerate quadratic forms $\mathbb{Q}[S_N]$ of the normalized sums ${S_N=N^{-1/2}(X_1+\cdots+X_N)}$ and show that, without any additional conditions, \[Δ_N\stackrel{\mathrm{def}}{=}\sup_x\bigl |\mathbf{P}\bigl\{\mathbb{Q}[…

Central Limit theorem for toric \kahler manifolds

Associated to the Bergman kernels of a polarized toric \kahler manifold $(M, ω, L, h)$ are sequences of measures $\{μ_k^z\}_{k=1}^{\infty}$ parametrized by the points $z \in M$. For each $z$ in the open orbit, we prove a central limit theorem for $μ_k^z$. The center of mass of $μ_k^z$ is the image of $z$ under the moment map; after re-centering at $0$ and dilating by $\sqrt{k}$, the re-normalized measure tends to a centered Gaussian whose variance is the Hessian of the \kahler potential at $z$. We further give a remainder estimate of Berry-Esseen type. The sequence $\{μ_k^z\}$ is generally not…

Regularity of solutions of the Stein equation and rates in the multivariate central limit theorem

Consider the multivariate Stein equation $Δf - x\cdot \nabla f = h(x) - E h(Z)$, where $Z$ is a standard $d$-dimensional Gaussian random vector, and let $f\_h$ be the solution given by Barbour's generator approach. We prove that, when $h$ is $α$-Hölder ($0<α\leq1$), all derivatives of order $2$ of $f\_h$ are $α$-Hölder {\it up to a $\log$ factor}; in particular they are $β$-Hölder for all $β\in (0, α)$, hereby improving existing regularity results on the solution of the multivariate Gaussian Stein equation. For $α=1$, the regularity we obtain is optimal, as shown by an example given by Raič \c…