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

2023/06/13 by Hollom, Lawrence, Portier, Julien
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2306.07811

Abstract

In this paper we prove that ℙ(|X| ≥ √(Var(X))) ≥ 7/32 for every finite Rademacher sum X, confirming a conjecture by Hitczenko and Kwapień from 1994, and improving upon results from Burkholder, Oleszkiewicz, and Dvořák and Klein. Moreover we fully determine the function f(y)= inf_X ℙ(|X| ≥ y√(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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