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

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, ${\sum }_{n\le x}f(n)n^{-\sigma }$, are positive for all $x\ge x_{\sigma }\ge 1$ in the regime $\sigma \to 1\mathrm{/}{2}^{+}$. In a previous paper by Heap, Zhao and the author, and by the author, when $0\le \sigma \le 1\mathrm{/}2$ this probability is zero. Here we give a positive lower bound for this probability depending on $x_{\sigma }$ that becomes large as $\sigma \to 1\mathrm{/}{2}^{+}$. 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.