On the supremum of the tails of normalized sums of independent Rademacher random variables

TL;DR

This paper disproves a longstanding conjecture regarding the maximum tail probabilities of normalized sums of independent Rademacher variables, contributing to the understanding of their probabilistic behavior.

Contribution

It provides a counterexample to the longstanding conjecture on the supremum of tail probabilities for Rademacher sums, challenging previous assumptions.

Findings

The original conjecture is false.

A related conjecture is also disproved.

Insights into the tail behavior of Rademacher sums.

Abstract

A well-known longstanding conjecture on the supremum of the tails of normalized sums of independent Rademacher random variables is disproved. A related conjecture, also recently disproved, is discussed.