Proof of Tomaszewski's conjecture on randomly signed sums

We prove the following conjecture, due to Tomaszewski (1986): Let X=∑i=1naixi, where ∑iai2=1 and each xi is a uniformly random sign. Then Pr⁡[|X|≤1]≥1/2. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for Rademacher sums.

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Vydáno v:Advances in mathematics (New York. 1965) Ročník 407; s. 108558
Hlavní autoři: Keller, Nathan, Klein, Ohad
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 08.10.2022
Témata:
Analysis of Boolean functions
Combinatorics
Probabilistic inequalities
Tail inequalities
Probabilistic inequalities
Analysis of Boolean functions
Tail inequalities
Combinatorics
ISSN:0001-8708, 1090-2082
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Popis
Shrnutí:We prove the following conjecture, due to Tomaszewski (1986): Let X=∑i=1naixi, where ∑iai2=1 and each xi is a uniformly random sign. Then Pr⁡[|X|≤1]≥1/2. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for Rademacher sums.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2022.108558