A Note on Distribution-Free Symmetrization Inequalities

Zhao Dong (), Jiange Li () and Wenbo V. Li
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Zhao Dong: Chinese Academy of Sciences
Jiange Li: University of Delaware
Wenbo V. Li: University of Delaware

Journal of Theoretical Probability, 2015, vol. 28, issue 3, 958-967

Abstract: Abstract Let $$X, Y$$ X , Y be two independent identically distributed (i.i.d.) random variables taking values from a separable Banach space . Given two measurable subsets , we establish distribution-free comparison inequalities between $$\mathbb {P}(X\pm Y \in F)$$ P ( X ± Y ∈ F ) and $$\mathbb {P}(X-Y\in K)$$ P ( X - Y ∈ K ) . These estimates are optimal for real random variables as well as when is equipped with the $$\Vert \cdot \Vert _\infty $$ ‖ · ‖ ∞ norm. Our approach for both problems extends techniques developed by Schultze and Weizsächer (Adv Math 208:672–679, 2007).

Keywords: Symmetrization inequalities; Distribution-free; Covering number; Kissing number; 52A40; 60C05; 60D05 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-014-0538-z

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