Large deviation upper bounds for sums of positively associated indicators

TL;DR

This paper derives exponential upper bounds for the probability that the sum of positively associated indicator variables is below a certain threshold, often outperforming existing inequalities like Janson's.

Contribution

It introduces new exponential bounds for sums of positively associated indicators, enabling comparison with the independent case and improving upon Janson's inequality in some scenarios.

Findings

Bounds are sometimes tighter than Janson's inequality.

Applicable to sums of positively associated indicators.

Provides a comparison framework with the independent case.

Abstract

We give exponential upper bounds for $P(S \le k)$, in particular $P(S=0)$, where $S$ is a sum of indicator random variables that are positively associated. These bounds allow, in particular, a comparison with the independent case. We give examples in which we compare with a famous exponential inequality for sums of correlated indicators, the Janson inequality. Here our bound sometimes proves to be superior to Janson's bound.