t-Wise Independence with Local Dependencies

TL;DR

This paper establishes a large deviation bound for sums of random variables with a dependency structure characterized by a graph with bounded chromatic number, where independent sets exhibit t-wise independence.

Contribution

It introduces a novel large deviation bound applicable to variables with local dependencies and t-wise independence within a graph-based dependency framework.

Findings

Proves a large deviation bound for dependent variables

Shows the bound applies to variables with bounded chromatic number

Extends analysis to t-wise independent collections within dependency graphs

Abstract

In this note we prove a large deviation bound on the sum of random variables with the following dependency structure: there is a dependency graph $G$ with a bounded chromatic number, in which each vertex represents a random variable. Variables that are represented by neighboring vertices may be arbitrarily dependent, but collections of variables that form an independent set in $G$ are $t$-wise independent.