Adrian P. C. Lim, Dejun Luo
We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if $A\subset\mathbb{R}^d$ is convex and the origin $0\in A$, then for any ball $B$ centered at the origin, it holds $γ_d(A\cap B)\geq γ_d(A)γ_d(B)$, where $γ_d$ is the standard Gaussian measure on $\mathbb{R}^d$. This generalizes Proposition 1 in [Arch. Rational Mech. Anal. 161 (2002), 257--269].
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Nonlocal operators related to nonsymmetric forms II: Harnack inequalities
本文针对与非对称型相关的非局部算子，证明了抛物和椭圆积分微分方程解的局部有界性及 Harnack 不等式，分别采用 De Giorgi 迭代和 Moser 迭代两种独立证明方法。
Nonsymmetric examples for Gaussian correlation inequalities
本文通过比较具有特定协方差结构与独立情形的标准高斯最大值方差，证明了一个非对称情形的凸/对数凹相关不等式，并推出独立同分布标准高斯最大值方差随 N 递减。
We establish transportation cost inequalities, with respect to the uniform and $L_2$-metric, on the path space of continuous functions, for laws of solutions of stochastic differential equations with reflections. We also consider the case of stochastic differential equations involving local times. Harnack inequalities for the associated semigroups are also established.
Lecture notes on martingale inequalities
本文介绍了几种证明鞅 $L^p$ 估计的技术，并展示了其在 Itô 积分和粗糙路径中的基本应用。
A note on oriented percolation with inhomogeneities and strict inequalities
针对会引入长程相关性的层状环境下定向渗流的相关问题，该注记将研究增加维度时定向渗流参数的严格下降问题，为相关研究提供工具支撑。
Hoeffding-type exponential inequalities are obtained for the distribution tails of canonical von Mises' statistics of arbitrary order based on samples from a stationary sequence of random variables satisfying the φ-mixing condition. The present paper weakens the restrictions on the coefficient φ() which are contained in the paper "Exponential inequalities for the distributions of canonical U- and V-statistics of dependent observations" by Borisov and Volodko (2009). At the same time, we correct the corresponding proof in this paper.
We derive inequalities for time-discrete and time-continuous martingales that are similar to the well-known Burkholder inequalities. For the time-discrete case arbitrary martingales in $L^p(Ω)$ are treated, whereas in the time-continuous case martingales defined by Itô integrals w.r.t. a multi-dimensional Wiener process are considered. The estimates for the time-discrete martingales are related to the more general results by I. Pinelis (1994) and are proved to be sharp by a different and more elementary proof for this special setting. Further, for time-continuous martingales the presented ineq…
Let $X, Y$ be two independent identically distributed (i.i.d.) random variables taking values from a separable Banach space $(\mathcal{X}, \|\cdot\|)$. Given two measurable subsets $F, K\subseteq\cal{X}$, we established distribution free comparison inequalities between $\mathbb{P}(X\pm Y \in F)$ and $\mathbb{P}(X-Y\in K)$. These estimates are optimal for real random variables as well as when $\mathcal{X}=\mathbb{R}^d$ is equipped with the $\|\cdot\|_\infty$ norm. Our approach for both problems extends techniques developed by Schultze and Weizsächer (2007).
A note on correlation inequalities for regular increasing families
本文研究了离散超立方体和高斯空间中单调事件与结构化阈值对象之间的定量相关不等式，证明了正则递增家族中存在线性阈值函数，其协方差下界为$ c \frac{\log n}{\sqrt{n}} $，并扩展到高斯空间中的半空间。
Some notes on moment inequalities for heavy-tailed distributions
本文研究重尾（尤其 Pareto 型）分布的矩与尾部关系，在正则性条件下验证结论锐性，并给出任意阶 $d$ 多项式混沌的集中界。
A note on the improved sparse Hanson-Wright inequalities
本文为稀疏 α-次指数随机向量的二次型建立并改进 Hanson-Wright 不等式，在 α≤1 时得到典型模型下的最优界，并展示其在协方差估计、低秩逼近和集中不等式中的应用。
We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusions with suitable drifts and diffusions. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.
In 1991, Persi Diaconis and Daniel Stroock obtained two canonical path bounds on the second largest eigenvalue for simple random walk on a connected graph, the Poincaré and Cheeger bounds, and they raised the question as to whether the Poincaré bound is always superior. In this paper, we present some background on these issues, provide an example where Cheeger beats Poincaré, establish some sufficient conditions on the canonical paths for the Poincaré bound to triumph, and show that there is always a choice of paths for which this happens.
We present some classical and weighted Poincaré inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors for a class of spherically symmetric probability measures in dimension larger than 2. Our strategy is based on two main ingredients: on the one hand, the optimal constant in the desired weighted Poincaré inequality has to be rewritten as the spectral gap of a convenient Markovian diffusion operator, and on the other hand we use a recent result given by the two first authors, which allows to estimate pr…
We prove extensions of classical concentration inequalities for random variables which have $α$-subexponential tail decay for any $α\in (0,2]$. This includes Hanson--Wright type and convex concentration inequalities. We also provide some applications of these results. This includes uniform Hanson--Wright inequalities and concentration results for simple random tensors in the spirit of previous work by Klochkov--Zhivotovskiy and Vershynin.
The aim of this paper is to prove an improved version of the bounded differences inequality for matrix valued functions, by developing the methods of Mackey et al.: "Matrix Concentration Inequalities via the Method of Exchangeable Pairs". Along the way, we prove new trace inequalities for the matrix exponential.
This note presents families of inequalities for the Gaussian measure of convex sets which extend the recently proven Gaussian correlation inequality in various directions.
The aim of this note is to give some Burkholder-Davis-Gundy type inequalities which are valid for the Ito stochastic integral with respect to Banach valued Levy noise.
This note deals with the question: what remains of the Burkholder-Davis-Gundy inequalities when stopping times $T$ are replaced by arbitrary random times $ρ$? We prove that these inequalities still hold when $T$ is a pseudo-stopping time and never holds for ends of predictable sets.
This note is concerned with lower tail estimates for product measures. Some improved deviation inequalities are obtained for functions satisfying some regularity and monotonicity assumptions. The arguments are based on semigroup interpolation together with Harris's negative association inequality and hypercontractive estimates.
This note is concerned with concentration inequalities for extrema of stationary Gaussian processes. It provides non-asymptotic tail inequalities which fully reflect the fluctuation rate, and as such improve upon standard Gaussian concentration. The arguments rely on the hypercontractive approach developed by Chatterjee for superconcentration variance bounds. Some statistical illustrations complete the exposition.
This note reviews the studies of the last decades emphasizing a common principle based on entropy, logarithmic Sobolev inequality and hypercontractivity, behind four most celebrated inequalities by M. Talagrand: the convex distance inequality, the $\mathrm {L}^1$-$\mathrm {L}^2$ variance inequality, the quadratic transportation cost inequality, and the inequality on the supremum of empirical processes.
In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.
In this note we establish several inequalities and monotonicity properties for the free energy of directed polymers under certain stochastic orders: the usual stochastic order, the Laplace transform order and the convex order. For the latter our results cover also many classical disordered systems.
We consider a Boolean model $Z$ driven by a Poisson particle process $η$ on a metric space $\mathbb{Y}$. We study the random variable $ρ(Z)$, where $ρ$ is a (deterministic) measure on $\mathbb{Y}$. Due to the interaction of overlapping particles, the distribution of $ρ(Z)$ cannot be described explicitly. In this note we derive concentration inequalities for $ρ(Z)$. To this end we first prove two concentration inequalities for functions of a Poisson process on a general phase space.
In this note we prove a weighted version of the Khintchine inequalities.
In this note we consider inequalities involving the error function $ϕ$. Our methodes give new proofs of some known inequalities of Komatsu, and of Szarek and Werner, and also produce two families of inequalities that give upper and lower bounds for $ϕ$. Moreover the continued fractions expansion of $ϕ$ is obtained.
We study the weighted Poincaré constant $C(p,w)$ of a probability density $p$ with weight function $w$ using integration methods inspired by Stein's method. We obtain a new version of the Chen-Wang variational formula which, as a byproduct, yields simple upper and lower bounds on $C(p,w)$ in terms of the so-called Stein kernel of $p$. We also iterate these variational formulas so as to build sequences of nested intervals containing the Poincaré constant, sequences of functions converging to said constant, as well as sequences of functions converging to the solutions of the corresponding spectr…
A note on concentration inequalities for the overlapped batch mean variance estimators for Markov chains
本文借助鞅分解方法，为一致几何遍历马尔可夫链渐近方差的 OBM 估计量建立集中不等式，给出误差 p 阶矩随 p 与混合时间变化的显式界。
We prove a new Bernstein-type inequality for the log-likelihood function of Bernoulli variables. In contrast to classical Bernstein's inequality and Hoeffding's inequality when applied to the log-likelihood, the new bound is independent of the parameters of the Bernoulli variables and therefore does not blow up as the parameters approach 0 or 1. The new inequality strengthens certain theoretical results on likelihood-based methods for community detection in networks and can be applied to other likelihood-based methods for binary data.
Isoperimetric inequalities in high-dimensional convex sets
本讲义笔记围绕Bourgain切片问题及Kannan、Lovasz与Simonovits提出的KLS等周猜想的近期研究进展展开阐述。
Assorted inequalities for pattern occurrences
本文在模式包含背景下，通过 FKG 不等式与 Shearer 引理的基本应用，建立了若干新的不等式。
The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $\R^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.
In this note we introduce and prove local and potential independent transportation, Log-Sobolev and HWI inequalities in one dimensional free probability on compact intervals which are sharp. We recover using this approach a free transportation inequality on the whole real line which was put forward recently by M. Maida and E. Maurel-Segala.
This note contains sufficient conditions for the probability density function of an arbitrary continuous univariate distribution, supported on $(0,\infty),$ such that the corresponding Mills ratio to be reciprocally convex (concave). To illustrate the applications of the main results, the reciprocal convexity (concavity) of Mills ratio of the gamma distribution is discussed in details.
Various New Inequalities for Beta Distributions
该研究针对 Beta 分布给出尾不等式、Hoeffding 与 Bernstein 型指数不等式、高斯不等式及近似等新结果。
On the Missing Factor in Some Concentration Inequalities for Martingales
本文针对有界增量鞅的集中不等式，恢复了Freedman型结果中缺失的因子并达到近最优，同时改进了独立随机变量函数的集中不等式。
The note is devoted to estimates for convolutions appearing in some class of stochastic Volterra equations. Two maximal inequalities and exponential tail estimate are proved by the fractional method of infinite dimensional stochastic calculus. The paper extends on non-semigroup case some results obtained earlier for semigroups.
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