Super-Gaussian Directions of Random Vectors

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.