{"ID":5346710,"CreatedAt":"2026-06-30T04:09:55.830587294Z","UpdatedAt":"2026-07-02T14:44:57.46949413Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.30411","arxiv_id":"2606.30411","title":"Notes on constants for maxima of Rademacher averages","abstract":"Let $ε_{ij}, i,j\\geq 1$ be independent Rademacher variables. We prove \\begin{equation*} \\mathbb{E} \\max_{1\\leq j\\leq p}\\left|\\frac{1}{n}\\sum_{i=1}^nε_{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.","short_abstract":"Let $ε_{ij}, i,j\\geq 1$ be independent Rademacher variables. We prove \\begin{equation*} \\mathbb{E} \\max_{1\\leq j\\leq p}\\left|\\frac{1}{n}\\sum_{i=1}^nε_{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)=(...","url_abs":"https://arxiv.org/abs/2606.30411","url_pdf":"https://arxiv.org/pdf/2606.30411v1","authors":"[\"Woonyoung Chang\"]","published":"2026-06-29T14:58:24Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\"]","methods":"[]","has_code":false}