Berry-Esseen theorem and local limit theorem for non uniformly expanding maps

An explicit Berry-Esséen bound for uniformly expanding maps on the interval

For uniformly expanding maps on the interval, analogous versions of the Berry-Esséen theorem are known but only with an unexplicit upper bound in $O(1/\sqrt{n})$ without any constants being specified. In this paper, we use the recent complex cone technique to prove an explicit Berry-Esséen estimate with a reasonable constant for these maps. Our method is not limited to maps on the interval however and should apply to many situations.

弱依赖随机变量的非均匀Berry-Esseen定理

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。

Berry-Esseen bound and Local Limit Theorem for the coefficients of products of random matrices

Let $μ$ be a probability measure on $\text{GL}_d(\mathbb{R})$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law $μ$. Under the assumptions that $μ$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry-Esseen bound with the optimal rate $O(1/\sqrt n)$ and a general Local Limit Theorem for the coefficients of $S_n$.

关于时间依赖扩展动力系统多变量Berry-Esséen定理

A multivariate Berry--Esseen theorem for time-dependent expanding dynamical systems

本文利用Stein方法，在非平稳过程中推导多变量中心极限定理的Berry-Esséen误差界，针对强混合基变换的随机动力系统，得出协方差矩阵线性增长时误差为O(N^{-1/2})的结论。

Berry-Esseen type estimate and return sequence for parabolic iteration in the upper half-plane

Two different aspects of parabolic iteration in the complex upper half-plane are considered here. First, from a noncommutative probability perspective, a Berry-Esseen type estimate for the convergence speed of the monotone central limit theorem is proved. Secondly, if the underlying measure in this central limit process is singular to the Lebesgue measure on the real line, then the iteration is shown to be an infinite-measure preserving dynamical system that has a regularly varying return sequence of index 1/2.

Berry-Esseen bounds with targets and Local Limit Theorems for products of random matrices

Let $μ$ be a probability measure on $\text{GL}_d(\mathbb R)$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$'s are i.i.d.'s with law $μ$. We study statistical properties of random variables of the form $$σ(S_n,x) + u(S_n x),$$ where $x \in \mathbb P^{d-1}$, $σ$ is the norm cocycle and $u$ belongs to a class of admissible functions on $\mathbb P^{d-1}$ with values in $\mathbb R \cup \{\pm \infty\}$. Assuming that $μ$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry-Esseen bounds and the Lo…