On a Conjecture of Feige for Discrete Log-Concave Distributions

TL;DR

This paper investigates Feige's conjecture for sums of independent discrete log-concave random variables, proving that the probability bound of 1/e holds in this class, thus strengthening the conjecture's validity.

Contribution

The paper proves that Feige's conjecture bound of 1/e applies to sums of independent discrete log-concave variables, extending the conjecture's scope.

Findings

Feige's bound of 1/e holds for discrete log-concave distributions.

The result applies to variables with arbitrary expectations.

Strengthens the conjecture's validity for a broader class of distributions.

Abstract

A remarkable conjecture of Feige (2006) asserts that for any collection of $n$ independent non-negative random variables $X_1, X_2, \dots, X_n$, each with expectation at most $1$, $$ \mathbb{P}(X < \mathbb{E}[X] + 1) \geq \frac{1}{e}, $$ where $X = \sum_{i=1}^n X_i$. In this paper, we investigate this conjecture for the class of discrete log-concave probability distributions and we prove a strengthened version. More specifically, we show that the conjectured bound $1/e$ holds when $X_i$'s are independent discrete log-concave with arbitrary expectation.