Sharp large deviation probabilities for sums of independent bounded   random variables
  We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving a one-term asymptotic expansion for large deviations. This result can also be regarded as sharp large deviations of types of Cram\'er and Bahadur-Ranga Rao. 
Xiequan Fan, Ion Grama and Quansheng Liu
math.PR