Extreme expectations of Bernoulli convolutions given their first few moments are attained at shifted convolutions of as few binomials

Abstract

A result of Chebyshev (1864) and Hoeffding1956}, on bounding an expectation of a given function with respect to a Bernoulli convolution (also called Poisson binomial law, or law of the number of successes in independent trials) with any given first moment, is here generalised to the case of any given first few moments, as indicated in the title. A nonprobabilistic, and perhaps more obvious, reformulation is: Every permutation invariant and separately affine-linear function of $n$ real variables $x_i\in[a,b]$ assumes its extremal values given the power sums $\sum_{i=1}^nx_i^1,\ldots, \sum_{i=1}^nx_i^r$ at vectors $x$ with at most $r$ coordinate values different from $a$ and $b$.