Electronic Journal of Probability

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

Iosif Pinelis

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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.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 20, 19 pp.

Received: 11 August 2015
Accepted: 25 February 2016
First available in Project Euclid: 11 March 2016

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G42: Martingales with discrete parameter 60G48: Generalizations of martingales

probability inequalities sums of random variables submartingales martingales upper bounds generalized moments

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Pinelis, Iosif. Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables. Electron. J. Probab. 21 (2016), paper no. 20, 19 pp. doi:10.1214/16-EJP4474. https://projecteuclid.org/euclid.ejp/1457706456

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