Small Deviations of Sums of Independent Random Variables

TL;DR

This paper improves lower bounds on the probability that the sum of independent nonnegative random variables deviates slightly above its expectation, narrowing the gap towards the conjectured optimal probability.

Contribution

The paper refines the lower bound on deviation probability by analyzing the first four moments, achieving approximately 0.14, closer to the conjectured 1/e.

Findings

Improved the lower bound to approximately 0.14

Analyzed first four moments for tighter bounds

Conjectured bound is around 0.368, still unproven

Abstract

A well-known discovery of Feige's is the following: Let $X_1, \ldots, X_n$ be nonnegative independent random variables, with $\mathbb{E}[X_i] \leq 1 \;\forall i$, and let $X = \sum_{i=1}^n X_i$. Then for any $n$, \[\Pr[X < \mathbb{E}[X] + 1] \geq \alpha > 0,\] for some $\alpha \geq 1/13$. This bound was later improved to $1/8$ by He, Zhang, and Zhang. By a finer consideration of the first four moments, we further improve the bound to approximately $.14$. The conjectured true bound is $1/e \simeq .368$, so there is still (possibly) quite a gap left to fill.