Multivariate CLT follows from strong Rayleigh property

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…

Stein方法与经典紧群上幂迹的多元CLT

Stein's method and the multivariate CLT for traces of powers on the classical compact groups

设$M_n$是酉群、特殊正交群或酉辛群的随机元，按Haar测度分布.根据Diaconis和Shahshahani的经典结果，对于大矩阵大小的$n$，向量$ (\on{Tr}(M_n), \on{Tr}(M_n^2),..., \on{Tr}(M_n^d))$倾向于独立（实或复）高斯随机变量的向量。最近，Jason Fulman已经证明，对于单功率$j$（其可以与$n$一起增长），可以通过Stein的可交换对方法获得收敛速度结果。本文将Fulman的结果推广到幂迹全向量的多元中心极限定理。

Lévy 过程的多变量中心极限定理：无矩假设下的收敛速率

Multivariate CLT for Lévy processes: convergence rates without moment assumptions

本文证明 Lévy 过程范数的二阶矩与凸距离可积性等价，并给出标准重标度下 $(2+\log)$-矩的充要条件，进而推导出无矩假设时收敛速率界的限制。

临界点的多元中心极限定理

Multivariable CLT for critical points

针对非各向同性高斯随机场，证明了所有指标临界点数量在水平集上的多元中心极限定理，并利用GOE矩阵特征值和Laplace方法处理非退化情形。

Randomized limit theorems for stationary ergodic random processes and fields

We consider "randomized" statistics constructed by using a finite number of observations a random field at randomly chosen points. We generalize the invariance principle (the functional CLT), the Glivenko--Cantelli theorem, the theorem about convergence to the Brownian bridge and the Kolmogorov theorem about the limit distribution of the empirical distribution function, as well as an improved version of the CLT in A. Tempelman, Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89-105. The randomized ap…

Multivariate approximation in total variation, I: equilibrium distributions of Markov jump processes

For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein--Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein's method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimens…

弱相依随机场的漂移集体积的中心极限定理

Central limit theorems for the excursion set volumes of weakly dependent random fields

证明了$\mathbb{R}^d$上平稳拟相伴随机场的漂移集体积的多元中心极限定理。特别注意高斯和散粒噪声领域。给出了极限分布的协方差阵的计算公式。还考虑了CLT的统计版本。文中还对一些数值结果进行了讨论。

高维随机效应模型中线性谱统计量的中心极限定理

CLT for Linear Spectral Statistics in High-Dimensional Random Effects Models

本文研究多层级方差分量样本协方差矩阵的线性谱统计量，在高维比例增长情形下证明其中心极限定理，并给出可数值计算的均值和协方差，为数量遗传学中嵌套随机效应模型的高维协方差估计提供理论基础。

高维和低维随机几何图的熵

Entropy of Random Geometric Graphs in High and Low Dimensions

本文利用多元中心极限定理研究了高维极限下随机几何图在立方体和环面上的分布，发现均匀分布或软RGGs收敛于ER系综，而非均匀分布或峰度大于1时最大熵较低，并通过Edgeworth修正推导了Shannon熵的次领头项，同时计算了低维情形的熵。

The Kadison-Singer Problem for Strongly Rayleigh Measures and Applications to Asymmetric TSP

Marcus, Spielman, and Srivastava in their seminal work \cite{MSS13} resolved the Kadison-Singer conjecture by proving that for any set of finitely supported independently distributed random vectors $v_1,\dots, v_n$ which have "small" expected squared norm and are in isotropic position (in expectation), there is a positive probability that the sum $\sum v_i v_i^\intercal$ has small spectral norm. Their proof crucially employs real stability of polynomials which is the natural generalization of real-rootedness to multivariate polynomials. Strongly Rayleigh distributions are families of probabili…

Multi-Normex Distributions for the Sum of Random Vectors. Rates of Convergence

We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called 'normex' approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named $d$-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the mu…

Chatterjee秩相关的极限定理

Limit theorems of Chatterjee's rank correlation

本文为一般随机变量对建立了Chatterjee秩相关的渐近正态性，证明其渐近方差一致有界于36且存在一致方差估计量，并将结果推广到Azadkia-Chatterjee图相关系数。