Deviation inequality for monotonic Boolean functions with application to a number of k-cycles in a random graph

Ville不等式向单调下界与阈值的推广

A Generalisation of Ville's Inequality to Monotonic Lower Bounds and Thresholds

该研究将经典Ville不等式的常数下界与阈值推广为单调曲线，给出保持在递减下界曲线上方的上鞅超过递增阈值曲线的概率紧界，并据此推导出有限时间版本重对数律。

Extended de Finetti theorems for boolean independence and monotone independence

We construct several new spaces of quantum sequences and their quantum families of maps in sense of Sołtan. Then, we introduce noncommutative distributional symmetries associated with these quantum maps and study simple relations between them. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by inte…

Stochastic Gronwall's inequality in random time horizon and its application to BSDE

In this paper, we introduce and prove a stochastic Gronwall's inequality in (unbounded) random time horizon. As an application, we prove a comparison theorem for backward stochastic differential equation (BSDE for short) with random terminal time under stochastic monotonicity condition.

Concentration and Moment Inequalities for Polynomials of Independent Random Variables

In this work we design a general method for proving moment inequalities for polynomials of independent random variables. Our method works for a wide range of random variables including Gaussian, Boolean, exponential, Poisson and many others. We apply our method to derive general concentration inequalities for polynomials of independent random variables. We show that our method implies concentration inequalities for some previously open problems, e.g. permanent of a random symmetric matrices. We show that our concentration inequality is stronger than the well-known concentration inequality due …

反极布尔函数的黑林格猜想

The Hellinger Conjecture for Antipodal Boolean Functions

该研究证明反极布尔函数的黑林格猜想，通过格拉姆矩阵核范数等方法将重叠度最小界扩展到任意独立二元输入信道，为信息论与测试提供相关界。

Convex Influences

We introduce a new notion of influence for symmetric convex sets over Gaussian space, which we term "convex influence". We show that this new notion of influence shares many of the familiar properties of influences of variables for monotone Boolean functions $f: \{\pm1\}^n \to \{\pm1\}.$ Our main results for convex influences give Gaussian space analogues of many important results on influences for monotone Boolean functions. These include (robust) characterizations of extremal functions, the Poincaré inequality, the Kahn-Kalai-Linial theorem, a sharp threshold theorem of Kalai, a stability ve…

Threshold for monotone symmetric properties through a logarithmic Sobolev inequality

Threshold phenomena are investigated using a general approach, following Talagrand [Ann. Probab. 22 (1994) 1576--1587] and Friedgut and Kalai [Proc. Amer. Math. Soc. 12 (1999) 1017--1054]. The general upper bound for the threshold width of symmetric monotone properties is improved. This follows from a new lower bound on the maximal influence of a variable on a Boolean function. The method of proof is based on a well-known logarithmic Sobolev inequality on $\{0,1\}^n$. This new bound is shown to be asymptotically optimal.

离散等周不等式：基于曲率的方法

Discrete Isoperimetric Inequalities via Curvature

本文提出基于曲率的半群框架，在弱依赖坐标测度下证明多种离散等周不等式，适用于Dobrushin唯一性区域的Ising模型。

A Sharp analog of Young's Inequality on $S^N$ and Related Entropy Inequalities

We prove a sharp analog of Young's inequality on $S^N$, and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions to optimizers in a monotonic manner. This strategy also works for the generalization of Young's inequality on $R^N$ to more than three functions, and leads to significant new information about the optimizers and the constants.

An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality

A certain spectrum, indexed by a\in[0,\infty], of upper bounds P_a(X;x) on the tail probability P(X\geq x), with P_0(X;x)=P(X\geq x) and P_\infty(X;x) being the best possible exponential upper bound on P(X\geq x), is shown to be stable and monotonic in a, x, and X, where x is a real number and X is a random variable. The bounds P_a(X;x) are optimal values in certain minimization problems. The corresponding spectrum, also indexed by a\in[0,\infty], of upper bounds Q_a(X;p) on the (1-p)-quantile of X is stable and monotonic in a, p, and X, with Q_0(X;p) equal the largest (1-p)-quantile of X. In …