An inequality for large deviation probabilities of sums of bounded i.i.d. random variables

We provide precise bounds for tail probabilities, say P{Mn > x}, of sums Mn of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed. © 2001 Plenum Publishing Corporation.