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Super-Gaussian Directions of Random Vectors

  • Bo’az Klartag
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.

Keywords

Random Vector Convex Body Universal Constant Tail Distribution Euclidean Ball 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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