Super-Gaussian directions of random vectors

TL;DR

This paper proves that in high dimensions, for any distribution of a random vector, there exists a direction along which the projection exhibits super-Gaussian tail behavior, with optimal dimension dependence.

Contribution

It establishes a universality property showing existence of a direction with super-Gaussian tails for any high-dimensional distribution, improving previous results on tail behavior.

Findings

Existence of a fixed direction with super-Gaussian tails in high dimensions.

Optimal dependence on the dimension n.

Improvement over previous work on tail behavior universality.

Abstract

We establish the following universality property in high dimensions: Let $X$ be a random vector with density in $\mathbb{R}^n$. The density function can be arbitrary. We show that there exists a fixed unit vector $\theta \in \mathbb{R}^n$ such that the random variable $Y = \langle X, \theta \rangle$ satisfies $$ \min \left \{ \mathbb{P}( Y \geq t M ), \mathbb{P}(Y \leq -tM) \right \} \geq c e^{-C t^2} \qquad \qquad \text{for all} \ 0 \leq t \leq \tilde{c} \sqrt{n}, $$ where $M > 0$ is any median of $|Y|$, i.e., $\min \{ \mathbb{P}( |Y| \geq M), \mathbb{P}( |Y| \leq M ) \} \geq 1/2$. Here, $c, \tilde{c}, C > 0$ are universal constants. The dependence on the dimension $n$ is optimal, up to universal constants, improving upon our previous work.