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A note on correlation inequalities for regular increasing families

Yiming Chen, Guozheng Dai

Abstract

This paper establishes quantitative correlation inequalities between monotone events and structured threshold objects in both the discrete cube and Gaussian space. We prove that for any increasing balanced family, there exists a linear threshold function yielding a covariance lower bound of $c \frac{\log n}{\sqrt{n}}$, and extend this principle to halfspaces in Gaussian space. These results verify the conjectures of Kalai, Keller, and Mossel regarding optimal correlation bounds for linear threshold functions and their Gaussian analogues.

A note on correlation inequalities for regular increasing families

Abstract

This paper establishes quantitative correlation inequalities between monotone events and structured threshold objects in both the discrete cube and Gaussian space. We prove that for any increasing balanced family, there exists a linear threshold function yielding a covariance lower bound of , and extend this principle to halfspaces in Gaussian space. These results verify the conjectures of Kalai, Keller, and Mossel regarding optimal correlation bounds for linear threshold functions and their Gaussian analogues.
Paper Structure (9 sections, 1 theorem, 145 equations)

This paper contains 9 sections, 1 theorem, 145 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorems \ref{['Theo_buchong']} and \ref{['Theo_main1']}
  3. Proof of Theorem \ref{['Theo_main_2']}
  4. Proof of Proposition \ref{['Prop_1']}
  5. Proof of Theorem \ref{['Theo_3']}
  6. Supplementary Proofs for Corollaries of Talagrand’s and KKM’s Results
  7. Gaussian Influence: An Interpretation
  8. Covariance Formulas for Gaussian Random Variables
  9. Auxiliary results

Key Result

Proposition 1

Let $w=(w_1, \cdots, w_n)$ be a vector in $\mathbb{R}^n$ such that $\Vert w\Vert_2=1$. Given $t\ge 1$, define the following LTF sets: Then, we have

Theorems & Definitions (12)

  • Proposition 1
  • proof : Proof of Theorem \ref{['Theo_buchong']}
  • proof : Proof of Theorem \ref{['Theo_main1']}
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['Theo_main_2']}
  • proof : Proof of Proposition \ref{['Prop_1']}
  • proof
  • proof
  • ...and 2 more