Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables

Abstract

Exact upper bounds on the generalized moments $\operatorname{\mathsf {E}} f(S_n)$ of sums $S_n$ of independent nonnegative random variables $X_i$ for certain classes $\mathcal{F} $ of nonincreasing functions $f$ are given in terms of (the sums of) the first two moments of the $X_i$’s. These bounds are of the form $\operatorname{\mathsf {E}} f(\eta )$, where the random variable $\eta $ is either binomial or Poisson depending on whether $n$ is fixed or not. The classes $\mathcal{F} $ contain, and are much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities $\operatorname{\mathsf {P}} (S_n\le x)$ are presented, for any real $x$. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments $\operatorname{\mathsf {E}} \exp \{hS_n\}$ for $h<0$, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.