Bounding probability of small deviation on sum of independent random variables: Combination of moment approach and Berry-Esseen theorem

Abstract: In the context of bounding probability of small deviation, there are limited general tools. However, such bounds have been widely applied in graph theory and inventory management. We introduce a common approach to substantially sharpen such inequality bounds by combining the semidefinite optimization approach of moments problem and the Berry-Esseen theorem. As an application, we improve the lower bound of Feige's conjecture from 0.14 to 0.1798.