On the positivity of some weighted partial sums of a random multiplicative function

2024/08/28 by Marco Aymone, Aymone, Marco · 2 citations
Decision Sciences · Mathematics · #Probability and Risk Models #Mathematical Approximation and Integration #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2408.15589

Abstract

Inspired by the papers by Angelo and Xu, Q.J Math., 74, pp. 767-777, and improvements by Kerr and Klurman, arXiv:2211.05540, we study the probability that the weighted sums of a Rademacher random multiplicative function, ∑_{n≤ x}f(n)n^-σ, are positive for all x≥ x_σ≥ 1 in the regime σ→1/2⁺. In a previous paper by Heap, Zhao and the author, and by the author, when 0≤ σ≤ 1/2 this probability is zero. Here we give a positive lower bound for this probability depending on x_σ that becomes large as σ→1/2⁺. The main inputs in our proofs are a maximal inequality based in relatively high moments for these partial sums combined with a Bonami--Halász's moment inequality, and also explicit estimates for the partial sums of non-negative multiplicative functions.

Cited by

• Positivity of partial sums of a random multiplicative function and corresponding problems for the Legendre symbol
• Weighted partial sums of a random multiplicative function and their positivity

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