Super-Gaussian directions of random vectors

Abstract

We establish the following universality property in high dimensions: Let X be a random vector with density in ℝn. The density function can be arbitrary. We show that there exists a fixed unit vector θ ∈ ℝn such that the random variable Y = 〈X, θ〉 satisfies (Formula presented) where M > 0 is any median of |Y|, i.e., min (Formula presented). Here, c, c,C > 0 are universal constants. The dependence on the dimension n is optimal, up to universal constants, improving upon our previous work.