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arXiv · 1301.0019

Small ball probability, Inverse theorems, and applications

Hoi H. Nguyen·Van H. Vu
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Abstract

Let $ξ$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 ξ_1 + ... + a_n ξ_n $$ where $ξ_i$ are iid copies of $ξ$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.

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Hoi H. Nguyen, Van H. Vu. 2012-12-31. Small ball probability, Inverse theorems, and applications. https://arxiv.org/abs/1301.0019

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