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We obtain some optimal inequalities on tail probabilities for sums ofindependent bounded random variables. Our main result completes an upper boundon tail probabilities due to Talagrand by giving a one-term asymptoticexpansion for large deviations. This result can also be regarded as sharp largedeviations of types of Cram\'er and Bahadur-Ranga Rao.

Sharp large deviation probabilities for sums of independent bounded random variables