On a conjecture concerning the sum of independent Rademacher random variables

TL;DR

This paper proves a sharp lower bound that at least half of the probability mass of a sum of independent Rademacher variables lies within one standard deviation, improving upon classical inequalities and with applications in sampling and random walks.

Contribution

It establishes a sharp lower bound for the probability mass within one standard deviation for sums of Rademacher variables, confirming a longstanding conjecture.

Findings

At least 50% of the probability mass is within one standard deviation.

The bound is sharp and improves upon Chebyshev's inequality.

Applications in finite sampling and random walk theory.

Abstract

It is shown that at least 50% of the probability mass of a sum of independent Rademacher random variables is within one standard deviation from its mean. This lower bound is sharp, it is much better than for instance the bound that can be obtained from application of the Chebishev inequality and the bound will have nice applications in finite sampling theory and in random walk theory. This old conjecture is of interest in itself, but has also an appealing reformulation in probability theory and in geometry.