Some Explorations on Two Conjectures About Rademacher Sequences

• 作者: Ze-chun HU Guo-lie LAN Wei SUN
• 来源期刊: 应用数学学报（英文版）
• 年，卷（期）: 2021,（1）
• 页码范围: 1-16
• 页数: 16页
• 语种: 英文

摘要

In this paper,we explore two conjectures about Rademacher sequences.Let(εi)be a Rademacher sequence,i.e.,a sequence of independent {-1,1}-valued symmetric random variables.Set Sn = a1ε1+γ… +anεn for a =(a1,…,an)∈ Rn.The first conjecture says that P(|Sn| ≤ ||a||)≥ 1/2 for all a∈Rn and n E N.The second conjecture says that P(|Sn| ≥ ||a||)≥ 3/72 for all a E Rn and n E N.Regarding the first conjecture,we present several new equivalent formulations.These include a topological view,a combinatorial version and a strengthened version of the conjecture.Regarding the second conjecture,we prove that it holds true when n ≤ 7.