On the accuracy in a combinatorial central limit theorem: the characteristic function method

Couplings and Matchings: Combinatorial notes on Strassen's theorem

Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall's marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory, in the sense that these results can easily be derived from each other. In this paper it is shown that this equivalence extends to a finite version of Strassen's theorem, a celebrated result on couplings of probability measures. Though this equivalence is known, probabilistic or combinatorial proofs of this fact are lacking. A novel combinatorial lemma will be…

经典紧群中局部线性统计量的中心极限定理及相关的组合恒等式

Central Limit Theorem for local linear statistics in classical compact groups and related combinatorial identities

本文讨论了来自经典紧群的随机矩阵的整体和局部线性统计量的CLT。我们的证明的主要部分是某些组合身份的精神，由卡茨和Spohn的作品。

Combinatorial considerations on the invariant measure of a stochastic matrix

The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit representation of the invariant measure of a stochastic matrix. In this note, we given a simple and purely combinatorial proof of the Markov tree theorem. In the symmetric case of detailed balance, the statement and the proof simplifies even more.

组合CLT中余项的Esseen型界

Esseen type bounds of the remainder in a combinatorial CLT

对于无三阶矩的独立随机变量，我们导出了组合中心极限定理中余项的Esseen型界.

一个鲁棒的定量局部中心极限定理及其在计数组合学和随机组合结构中的应用

A robust quantitative local central limit theorem with applications to enumerative combinatorics and random combinatorial structures

在理解大型随机组合结构中的一个有用的启发是Arratia-Tavare原理，该原理描述了使用独立随机变量对组件大小的联合分布的近似。该原则概述了条件下，全变差距离真正的联合分布和近似应该是小的，并成功地利用皮特尔的情况下，整数分区和集分区。我们提供了充分条件，这一原则是真实的，在一般情况下，有效的某些离散概率分布是$\textit{perturbed log-concave}$，通过一个定量的本地中心极限定理。然后，我们用它来推广一些经典的渐近统计组合理论，以及断言一些新的。

部分和乘积的中心极限定理与几乎几乎必然中心极限定理

Central limit theorem and almost sure central limit theorem for the product of some partial sums

本文给出了独立同分布随机变量部分和乘积的中心极限定理和几乎必然中心极限定理。

Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices

An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $η\geq 0$, $\Pr(s_n(M_n) \leq η) \lesssim n^{C}η+ \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work…

从经典到现代的中心极限定理

From classical to modern central limit theorems

本文回顾了从De Moivre发现正态分布和经典中心极限定理，到鞅中心极限定理的历史，并介绍了非线性中心极限定理和非线性正态分布这一新研究方向。

A combinatorial proof of Aldous-Broder theorem for general Markov chains

Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph $G$, but it is more general: given an irreducible and reversible Markov chain $M$ on $G$ started at $r$, the tree rooted at $r$ formed by the first entrance steps in each node (different from the root) has a probability proportional to $\prod_{e=(e^-,e^+)\in {\sf Edges}(t,r)} M_{e^{-},e^+}$, where the edges are directed toward $r$. In this paper we give proofs of Aldous-Broder theorem in the general case, where the kernel $M$ is irreducible but not assumed to be reversible (this g…

集合划分的新组合恒等式与有限自由概率论中的极限定理

New combinatorial identity for the set of partitions and limit theorems in finite free probability theory

本文通过建立集合划分的精细化组合恒等式，研究了有限自由概率论中的极限定理，给出了 Sakuma–Yoshida 定理的有限自由类比、有限自由乘法卷积的中心极限定理，并为 Kabluchko 的相关极限定理提供了新证明。

约束随机游动模型的组合方法

A combinatorial approach to a model of constrained random walkers

在文献[1]中，作者考虑了随机游动$(Z_{n,1},\ldots,Z_{n,K+1})\in \mathbb{Z}^{K+1}$，其约束条件是游动的每个坐标与下一个坐标的距离为1。证明了第一坐标的泛函中心极限定理，并给出了极限方差的显式表达式。在本文中，我们研究了这个模型的一个扩展版本的条件极值坐标是在某个固定的距离在每一个时间。证明了该随机游动的一个泛函中心极限定理。利用组合工具，我们给出了方差的精确公式，并与文献[1]中的公式进行了比较。

可逆凝聚-碎裂过程和随机组合结构：群数的渐近性

Reversible coagulation-fragmentation processes and random combinatorial structures:asymptotics for the number of groups

本文建立了由多项式增长率的参数函数给出的凝聚-破碎过程平衡点群数的中心极限定理。所得结果与服从对数条件的随机组合结构的结果进行了比较。

一个组合中心极限定理的误差界

On the error bound in a combinatorial central limit theorem

令$\mathbb{X}=\{X_{ij}: 1\le i,j\le n\}$是独立随机变量的$n\times n$阵列，其中$n\ge2$。令$π$是$\{1,2,\dots,n\}$的均匀随机置换，与$\mathbb{X}$无关，令$W=\sum_{i=1}^nX_{iπ(i)}$。假设$\mathbb{X}$被标准化，使得${\mathbb{E}}W=0,\operatorname {Var}(W)=1$ .我们证明了$W$的分布与标准正态分布之间的Kolmogorov距离是由$451\sum_{i,j=1}^n{\mathbb{E}}|X_{ij}|^3/n$限定的。我们的方法是通过Stein的方法交换对和使用的浓度不等式。