Some explorations on two conjectures about Rademacher sequences

2019/10/24 by Ze-Chun Hu, Hu, Ze-Chun, Guolie Lan +3
Mathematics · #60C05 #60G50 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #math.CO #math.PR #msc:60C05 #msc:60G50

paper · pdf · doi:10.48550/arxiv.1910.11312

19 pages

arxiv created 2019/10/24 · arxiv updated 2019/10/25

Abstract

In this paper, we explore two conjectures about Rademacher sequences. Let (ε_i) be a Rademacher sequence, i.e., a sequence of independent \-1,1\-valued symmetric random variables. Set S_n=a₁ε₁+⋯+a_nε_n for a=(a₁,…,a_n)∈ ℝⁿ. The first conjecture says that P ( |S_n |≤ ‖a‖ )≥(1)/(2) for all a∈ ℝⁿ and n∈ ℕ. The second conjecture says that P ( |S_n |≥‖a‖ )≥ (7)/(32) for all a∈ ℝⁿ and n∈ 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.

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