Abstract
Let \(X, Y\) 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 \(\mathbb {P}(X\pm Y \in F)\) and \(\mathbb {P}(X-Y\in K)\). 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).
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Acknowledgments
We are grateful to Dr. Mokshay Madiman for his valuable suggestions and comments in the preparation of this note.
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Prof. Wenbo V. Li is deceased.
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Dong, Z., Li, J. & Li, W.V. A Note on Distribution-Free Symmetrization Inequalities. J Theor Probab 28, 958–967 (2015). https://doi.org/10.1007/s10959-014-0538-z
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DOI: https://doi.org/10.1007/s10959-014-0538-z