H\"older-type inequalities and their applications to concentration and correlation bounds

TL;DR

This paper introduces a H"older-type inequality for dependent random variables based on the graph chromatic number, and applies it to derive bounds on concentration and correlation in weakly dependent systems.

Contribution

It extends H"older's inequality to dependent variables using graph chromatic numbers and demonstrates its utility in concentration and correlation bounds.

Findings

Derived a new inequality for dependent variables using graph coloring.

Applied the inequality to obtain concentration bounds.

Provided correlation bounds for weakly dependent variables.

Abstract

Let $Y_v, v\in V,$ be $[0,1]$-valued random variables having a dependency graph $G=(V,E)$. We show that \[ \mathbb{E}\left[\prod_{v\in V} Y_{v} \right] \leq \prod_{v\in V} \left\{ \mathbb{E}\left[Y_v^{\frac{\chi_b}{b}}\right] \right\}^{\frac{b}{\chi_b}}, \] where $\chi_b$ is the $b$-fold chromatic number of $G$. This inequality may be seen as a dependency-graph analogue of a generalised H\"older inequality, due to Helmut Finner. Additionally, we provide applications of H\"older-type inequalities to concentration and correlation bounds for sums of weakly dependent random variables.