Optimal Probability Inequalities for Random Walks related to Problems in Extremal Combinatorics

TL;DR

This paper establishes optimal probability inequalities for sums of symmetric random variables, improving existing bounds, and provides new proofs and extensions relevant to extremal combinatorics and probability theory.

Contribution

It proves an optimal probability inequality for sums of symmetric variables, improving Kwapień's inequality, and offers new proofs and extensions for related combinatorial problems.

Findings

Proved an exact inequality for symmetric sums with explicit optimal constants.

Improved upon Kwapień's inequality for Rademacher series.

Provided a new short proof of the Littlewood-Offord problem.

Abstract

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that \mathbb{P}{S_{n} \in A} \leq \mathbb{P}{cW_k \in A}, where A is either an interval of the form [x, \infty) or just a single point. The inequality is exact and the optimal values of c and k are given explicitly. It improves Kwapie\'n's inequality in the case of the Rademacher series. We also provide a new and very short proof of the Littlewood-Offord problem without using Sperner's Theorem. Finally, an extension to odd Lipschitz functions is given.