Sharp small-deviation inequalities for sums of independent nonnegative random variables

Abstract

Let $(X_1,\dots ,X_n)$ be independent nonnegative random variables with $\mathbb{E}{X}_i\le 1$, and write $S={\sum }_i{X}_i$. For $\hat{I}´>0$, we prove that \[ \mathbb{P}\left(S<\mathbb{E} S+δ\right)\ge b_{n,δ}, \] where $b_{n,\hat{I}´}=\hat{I}´(n/(n+\hat{I}´){)}^n$ for $0<\hat{I}´<1$ and $b_{n,\hat{I}´}=(1-1/(n+\hat{I}´){)}^n$ for $\hat{I}´\ge 1$. The bound is sharp for every $n$ and $\hat{I}´\ge 1$. In particular, since $b_{n,\hat{I}´}\ge e^{-1}$ for $\hat{I}´\ge 1$, our result proves Feige's conjecture [Feige, 2004] in the affirmative for $\hat{I}´\ge 1$. The proof is found by ChatGPT 5.6 Pro. It combines the exact Dirichlet calibration theorem of Vlassis and Thomas [Vlassis and Thomas, 2026], which resolves Gaffke's conjecture in statistics, with results in convex geometry including Grünbaum's centroid theorem [Grünbaum, 1960] and its generalization by Letwin and Yaskin [Letwin and Yaskin, 2024].