A Tail Bound for Read-k Families of Functions

TL;DR

This paper establishes a Chernoff-like large deviation bound for sums of dependent Boolean functions of independent variables, where each variable influences at most k of the functions, extending classical concentration inequalities.

Contribution

It introduces a new tail bound for read-k families of functions, generalizing Chernoff bounds to certain dependent structures.

Findings

Provides a Chernoff-like bound for read-k families

Extends concentration inequalities to dependent variables

Applicable to Boolean functions with limited influence

Abstract

We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,...,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,...,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,...,Y_r$.