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

TL;DR

This paper generalizes classical bounds on expectations of functions with respect to Bernoulli convolutions to include multiple moments, showing extremal values occur at shifted convolutions of few binomials, with implications for permutation-invariant functions.

Contribution

It extends Chebyshev and Hoeffding bounds to multiple moments and provides a nonprobabilistic reformulation involving extremal values of affine-linear functions.

Findings

Expectations are maximized at shifted convolutions of few binomials.

Extremal values of permutation-invariant functions occur at vectors with limited coordinate variation.

Generalization applies to bounding functions of Bernoulli convolutions using multiple moments.

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 $b$.