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 $\mathbb{R}^{n}$. 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

$$\displaystyle{ \min \left \{\mathbb{P}(Y \geq tM), \mathbb{P}(Y \leq -tM)\right \} \geq ce^{-Ct^{2} }\qquad \qquad \text{for all}\ 0 \leq t \leq \tilde{ c}\sqrt{n}, }$$

where M > 0 is any median of | Y | , i.e., $\min \{\mathbb{P}(\vert Y \vert \geq M), \mathbb{P}(\vert Y \vert \leq M)\} \geq 1/2$. 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.