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

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

2011-11-28

arxiv.org/abs/1111.6332v2

Mathematics

Probability

10 pages, submitted

Scientific paper

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.

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