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A Note on Distribution-Free Symmetrization Inequalities

Author

Listed:
  • Zhao Dong

    (Chinese Academy of Sciences)

  • Jiange Li

    (University of Delaware)

  • Wenbo V. Li

    (University of Delaware)

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).

Suggested Citation

  • Zhao Dong & Jiange Li & Wenbo V. Li, 2015. "A Note on Distribution-Free Symmetrization Inequalities," Journal of Theoretical Probability, Springer, vol. 28(3), pages 958-967, September.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:3:d:10.1007_s10959-014-0538-z
    DOI: 10.1007/s10959-014-0538-z
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