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

Weibo Fu, Yanjun Han, Guanyang Wang +3

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+Î…