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In the context of bounding probability of small deviation, there are limitedgeneral tools. However, such bounds have been widely applied in graph theoryand inventory management. We introduce a common approach to substantiallysharpen such inequality bounds by combining the semidefinite optimizationapproach of moments problem and the Berry-Esseen theorem. As an application, weimprove the lower bound of Feige's conjecture from 0.14 to 0.1798.

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