关于集中性的逻辑方法

关于随机对称矩阵特征值的集中性

On the concentration of eigenvalues of random symmetric matrices

我们证明了几个最大的（和最重要的）特征值的随机对称矩阵的各种非常强烈的集中。这种强烈的集中使我们能够以高精度计算这些特征值的平均值。我们的方法使用Talagrand不等式，是非常不同的标准方法。

结构化息票收集器覆盖时间的速度和集中度

Speed and concentration of the covering time for structured coupon collectors

设$V$是一个$n$ -集，$X$是一个随机变量，取值于$V$的幂集中.假设我们给出一个随机券序列$X_1, X_2, \ldots $，其中$X_i$是独立的随机变量，其分布由$X$给出。覆盖时间$T$是使得$\bigcup_{i=1}^tX_i=V$ . $T$的分布在组合概率的许多应用中是重要的，并且已经被广泛地研究。然而，文献几乎完全集中在假设$X$以某种方式对称和/或均匀的情况下。在本文中，我们研究了覆盖时间更一般的随机变量$X$，我们给出了一般的标准$T$急剧集中在其平均值，精确的工具来估计，平均值，以及例子$T$未能集中时，结构特性的分布$X$允许一个非常不同的行为$T$相对于对称/均匀的情况。

随机行列式的集中与永久估计量

Concentration of random determinants and permanent estimators

我们表明，随机独立（但不一定iid）的元素的矩阵的行列式的绝对值是强烈集中在其平均值。作为一个应用，我们证明了Godsil-Gutman和Barvinok估计的永久性的严格正矩阵的次指数逼近比率的高概率。

Concentration of maximum degree in random planar graphs

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $[n]:=\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $m/n\leq 1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values. In contrast, this is not true anymore in the dense regime, when $m/n>1$, where the maximum degree of $P(n,m)$ is not concentrated on any subset of $[n]$ with bounded size.

关于$G_{n,m}$独立数的两点集中性的注记

A note on Two-Point Concentration of the Independence Number of $G_{n,m}$

本研究证明了在边数满足$n^{5/4+ε}<m≤\binom{n}{2}$时随机图$G_{n,m}$的独立数集中在两个值上，明确了其与对应$G_{n,p}$模型在$n^{5/4+ε}<m<n^{4/3}$区间内的集中性差异。

A spectral characterization for concentration of the cover time

We prove that for a sequence of finite vertex-transitive graphs of increasing sizes, the cover times are asymptotically concentrated if and only if the product of the spectral-gap and the expected cover time diverges. In fact, we prove this for general reversible Markov chains under the much weaker assumption (than transitivity) that the maximal hitting time of a state is of the same order as the average hitting time.

一般随机有向图上平稳分布的集中性

Concentration of the Stationary Distribution on General Random Directed Graphs

我们考虑一个随机模型的有向图，其中一个弧被放置从一个顶点到另一个具有规定的概率，可能会有所不同，从弧到弧。本文利用扰动界和Mrsoff不等式，证明了随机图上的马尔可夫过程的平稳分布在较弱的条件下集中在“期望”过程的平稳分布附近.这些条件包括最小和最大的入度和出度之间的比率、平稳分布中最小和最大条目的比率以及转移阵的最小奇异值。最后，我们给出了我们的结果的应用程序的例子，如PageRank和G（n，p）的知名模型。

非常度随机正则图的色数的集中性

Random regular graphs of non-constant degree: Concentration of the chromatic number

在这项工作中，我们表明，以高概率的色数的随机正则图模型$\Gnd$为$d=o(n^{1/5})$的一个图形采样集中在两个连续的值，从而扩展了Achlioptas和摩尔以前的结果。这种集中现象与具有$p=\frac{d}{n}$的二项随机图模型$\Gnp$的集中现象非常相似。我们的证明主要基于Alon和Krivelevich的思想，他们证明了$\Gnp$的两点浓度结果，其中$p=n^{-δ}$为$δ>1/2$。用于得出这样一个结果的主要工具是仔细分析的边缘分布在$\Gnd$，既依赖于开关技术和边界的概率指数小事件的配置模型。

Concentration of Measure and Large Random Matrices with an application to Sample Covariance Matrices

The present work provides an original framework for random matrix analysis based on revisiting the concentration of measure theory from a probabilistic point of view. By providing various notions of vector concentration ($q$-exponential, linear, Lipschitz, convex), a set of elementary tools is laid out that allows for the immediate extension of classical results from random matrix theory involving random concentrated vectors in place of vectors with independent entries. These findings are exemplified here in the context of sample covariance matrices but find a large range of applications in st…

Concentration inequalities for sums of random variables, each having power bounded tails

In this work we present concentration inequalities for the sum $S_n$ of independent integer-valued not necessary indentically distributed random variables, where each variable has tail function that can be bounded by some power function with exponent $-α$. We show that when $0<α\leq 1$, then the sum does not have finite expectation, however, with high probability we have that $|S_n|=O\left(n^{1/α}\right)$. When $α>1$, then the sum $S_n$ is concentrated around its mean. Since the r.vs. that constitute the sum has tails, which can be bounded by some power function, it follows that results of thi…

可交换随机变量的Hoeffding型浓度界

Hoeffding-Type Concentration Bounds for Exchangeable Random Variables

针对可交换随机变量序列，提出基于de Finetti混合测度支撑中分布的最大和最小均值的Hoeffding型浓度不等式，填补了有限样本均值与总体均值之间的空白。

Concentration of the spectral norm of Erdős-Rényi random graphs

We present results on the concentration properties of the spectral norm $\|A_p\|$ of the adjacency matrix $A_p$ of an Erdős-Rényi random graph $G(n,p)$. First we consider the Erdős-Rényi random graph process and prove that $\|A_p\|$ is uniformly concentrated over the range $p\in [C\log n/n,1]$. The analysis is based on delocalization arguments, uniform laws of large numbers, together with the entropy method to prove concentration inequalities. As an application of our techniques we prove sharp sub-Gaussian moment inequalities for $\|A_p\|$ for all $p\in [c\log^3n/n,1]$ that improve the general…

随机矩阵：特征值的急剧集中

Random matrices: Sharp concentration of eigenvalues

令$W_n= \frac{1}{\sqrt n} M_n$是其条目具有消失的三阶矩的维格纳矩阵，归一化使得频谱集中在间隔$[-2,2]$中。我们证明了$N_I = N_I(W_n)$的一个浓度界，即$W_n$在一个区间$I$中的特征值的个数。我们的结果表明，$N_I$指数衰减的标准差最多$O(\log^{O(1)} n)$。这是最好的可能达到常数指数的对数项。作为推论，体特征值被局部化到宽度为$O(\log^{O(1)} n/n)$的区间;同样，这是最优的，直到指数。这些结果加强了Erdos，Yau和Yin（在三分之一消失的额外假设下）的最近结果

A Concentration of Measure and Random Matrix Approach to Large Dimensional Robust Statistics

This article studies the \emph{robust covariance matrix estimation} of a data collection $X = (x_1,\ldots,x_n)$ with $x_i = \sqrt τ_i z_i + m$, where $z_i \in \mathbb R^p$ is a \textit{concentrated vector} (e.g., an elliptical random vector), $m\in \mathbb R^p$ a deterministic signal and $τ_i\in \mathbb R$ a scalar perturbation of possibly large amplitude, under the assumption where both $n$ and $p$ are large. This estimator is defined as the fixed point of a function which we show is contracting for a so-called \textit{stable semi-metric}. We exploit this semi-metric along with concentration …