PROBABILITY BOUNDS ON THE SUM OF INDEPENDENT NONIDENTICALLY DISTRIBUTED BINOMIAL RANDOM VARIABLES.

The cumulative distribution of the sum of independent binomial random variables is investigated. After writing down exact expressions for these quantities, the authors develop a sequence of increasingly tight upper and lower bounds, given various characteristics of the underlying set of probabilities. The major tool in each case is a transformation of the probability set for which the cumulative distributions act as Lyapounov function. Their most sophisticated bounds, in which the first two cumulatives are given, are computed for a number of sets of probabilities and compared with familiar results in the literature. They are uniformly superior.