# Decoupling Maximal Inequalities

## TL;DR

This paper investigates how maximal inequalities behave under different dependence structures among non-negative random variables, showing that pairwise independence and certain negative dependencies still allow for effective bounds.

## Contribution

It demonstrates that pairwise independence and specific negative dependence conditions are sufficient for maximal inequalities to hold similarly to the independent case.

## Key findings

- Pairwise independence suffices for maximal inequalities to behave like the independent case.
- Negative dependence conditions can also ensure similar maximal inequality bounds.
- Violations of negative dependence can be tolerated if properly quantified.

## Abstract

A {\em maximal inequality} seeks to estimate $\mathbb{E}\max_i X_i$ in terms of properties of the $X_i$. When the latter are independent, the union bound (in its various guises) can yield tight upper bounds. If, however, the $X_i$ are strongly dependent, the estimates provided by the union bound will be rather loose. In this note, we show that for non-negative random variables, pairwise independence suffices for the maximal inequality to behave comparably to its independent version. The condition of pairwise independence may be relaxed to a kind of negative dependence, and even the latter admits violations -- provided these are properly quantified.