A note on the Hitczenko-Kwapien conjecture about a Rademacher sequence

2023/10/23 by Shi-Zhen Liu, Liu, Shi-Zhen, Ze-Yu Tao +3
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2310.14878

openalex publication_date 2023/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (ε_i) be a Rademacher sequence, i.e., a sequence of independent and identically distributed random variables satisfying P(ε_i=1)=P(ε_i=-1)=1/2. Set S_n=a₁ε₁+⋯+a_nε_n for a=(a₁,…,a_n)∈ ℝⁿ. The Hitczenko-Kwapien conjecture says that P(|S_n|≥‖a‖)≥ 7/32 for all a∈ ℝⁿ and n∈ ℕ. Up to now, we know that it holds when n≤ 7. In this note, we show that it holds when n=8.

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