论文标题
通过半群插值的定量相关性不平等
Quantitative Correlation Inequalities via Semigroup Interpolation
论文作者
论文摘要
文献中高维功能的大多数相关性不平等，例如fortuin-kasteleyn-吉尼伯（FKG）不平等和罗伊恩（Royen）著名的高斯相关性不平等，都是定性陈述，它们确定了某种类型的任何两种功能都具有非维持率相关性。在这项工作中，我们提供了一种通用方法，该方法可用于引导许多定性相关性不平等，以使产品空间上的功能成为定量陈述。该方法结合了有关功率序列的新的极端结果，该结果使用复杂分析证明，以及对产品空间功能的谐波分析。我们在几种不同的具体设置中实例化这种通用方法，以获得一系列新的和近乎最佳的定量相关性不平等，包括： $ \ bullet $ Royen著名的高斯相关性不平等的定量版本。 Royen（2014）证实了一个猜想，开放了40年，指出任何两个对称的凸集必须在任何中心的高斯分布下都无关。我们在两个凸组集合的2度HERMITE系数的矢量方面给出了相关性的下限，类似于Talagrand（1996）获得的单调布尔函数的相关性。 $ \ bullet $在任何有限的产品概率空间上用于单调函数的众所周知的FKG不等式的定量版本，概括了由$ \ {0,1 \}^n $限制的单调布尔函数的定量相关性。我们知道的唯一先前的概括是凯勒（Keller（2008，2009，2012）），它将talagrand的结果扩展到了$ \ {0,1 \}^n $的产品分布。我们还提供了两个不同的定量版本的FKG不等式，用于在连续域上$ [0,1]^n $，以回答Keller（2009）的问题。
Most correlation inequalities for high-dimensional functions in the literature, such as the Fortuin-Kasteleyn-Ginibre (FKG) inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have non-negative correlation. In this work we give a general approach that can be used to bootstrap many qualitative correlation inequalities for functions over product spaces into quantitative statements. The approach combines a new extremal result about power series, proved using complex analysis, with harmonic analysis of functions over product spaces. We instantiate this general approach in several different concrete settings to obtain a range of new and near-optimal quantitative correlation inequalities, including: $\bullet$ A quantitative version of Royen's celebrated Gaussian Correlation Inequality. Royen (2014) confirmed a conjecture, open for 40 years, stating that any two symmetric, convex sets must be non-negatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree-2 Hermite coefficients of the two convex sets, analogous to the correlation bound for monotone Boolean functions over $\{0,1\}^n$ obtained by Talagrand (1996). $\bullet$ A quantitative version of the well-known FKG inequality for monotone functions over any finite product probability space, generalizing the quantitative correlation bound for monotone Boolean functions over $\{0,1\}^n$ obtained by Talagrand (1996). The only prior generalization of which we are aware is due to Keller (2008, 2009, 2012), which extended Talagrand's result to product distributions over $\{0,1\}^n$. We also give two different quantitative versions of the FKG inequality for monotone functions over the continuous domain $[0,1]^n$, answering a question of Keller (2009).
