The paper characterizes the worst-case expected top-k norm of sample averages for heavy-tailed vectors up to universal constants under envelope moment conditions.
Notes on constants for maxima of Rademacher averages
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abstract
Let $\epsilon_{ij}, i,j\geq 1$ be independent Rademacher variables. We prove \begin{equation*} \mathbb{E} \max_{1\leq j\leq p}\left|\frac{1}{n}\sum_{i=1}^n\epsilon_{ij}\right| \geq \min\left\{\frac{255}{256},\frac{1}{\sqrt{2\log 2}}\sqrt{\frac{\log(2p)}{n}}\right\}. \end{equation*} The equality is attained, for instance, by $(n,p)=(2,1)$ and $(n,p)=(2,8).$ We also discuss the optimality of the numerical constants.
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Worst-Case Maximal Inequalities for Heavy-tailed Random Vectors
The paper characterizes the worst-case expected top-k norm of sample averages for heavy-tailed vectors up to universal constants under envelope moment conditions.