Improved Estimates for Binomial Tails

In a recent paper, Ferrante gave these estimates for binomial tails, P(Sn(x) ≥ na) ≤ Fn(a, x) exp(−nH (α, x)), in terms of the Kullback-Leibler divergence H(α, x) between Bernoulli random variables. Although the upper bound is asymptotically sharp in n for fixed values of a and x, the term Fn(a, x) goes to infinity when a approaches to x. In this note, we replace Fn(a, x) by a bounded term making use of the Berry-Esseen inequality. As an application, we show that Bernstein polynomials acting on locally affine functions exhibit an exponential rate of pointwise convergence, in contrast with the usual polynomial rates.