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arXiv · 2306.07811

Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem

Lawrence Hollom·Julien Portier
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Abstract

In this paper we prove that $\mathbb{P}(|X| \geq \sqrt{\text{Var}(X)}) \geq 7/32$ for every finite Rademacher sum $X$, confirming a conjecture by Hitczenko and Kwapie{\'n} from 1994, and improving upon results from Burkholder, Oleszkiewicz, and Dvo\v{r}\'ak and Klein. Moreover we fully determine the function $f(y)= \inf_X \mathbb{P}(|X| \geq y\sqrt{\text{Var}(X)})$ where the $\inf$ is taken over all finite Rademacher sums $X$, confirming a conjecture by Lowther and giving a partial answer to a question by Keller and Klein.

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Lawrence Hollom, Julien Portier. 2023-06-13. Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem. https://arxiv.org/abs/2306.07811

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