Super-Gaussian Directions of Random Vectors

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2169)

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.

Notes

Acknowledgements

I would like to thank Bo Berndtsson and Emanuel Milman for interesting discussions and for encouraging me to write this paper. Supported by a grant from the European Research Council.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsWeizmann Institute of ScienceRehovotIsrael
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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