Super-Gaussian Directions of Random Vectors

Research output: Chapter in Book/Report/Conference proceedingChapter


We establish the following universality property in high dimensions: Let X be a random vector with density in RnRn. The density function can be arbitrary. We show that there exists a fixed unit vector θ∈Rnθ∈Rn such that the random variable Y=⟨X,θ⟩Y=⟨X,θ⟩ satisfies

min{P(Y≥tM),P(Y≤−tM)}≥ce−Ct2for all 0≤t≤c~n−−√,min{P(Y≥tM),P(Y≤−tM)}≥ce−Ct2for all 0≤t≤c~n,

where M > 0 is any median of | Y | , i.e., min{P(|Y|≥M),P(|Y|≤M)}≥1/2min{P(|Y|≥M),P(|Y|≤M)}≥1/2. Here, c,c~,C>0c,c~,C>0 are universal constants. The dependence on the dimension n is optimal, up to universal constants, improving upon our previous work.

Original languageEnglish
Title of host publicationGeometric Aspects of Functional Analysis
Subtitle of host publicationIsrael Seminar (GAFA) 2014–2016
EditorsBo'az Klartag, Emanuel Milman
Place of PublicationCham, Switzerland
Number of pages25
ISBN (Electronic)9783319452821
StatePublished - 2017

Publication series

NameLecture Notes in Mathematics
ISSN (Print)0075-8434

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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