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

Inspired in 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, ₍ ₗf (n) n^-, are positive for all x x_ 1 in the regime 1/2^+. In a previous paper by the author, when 1/2 this probability is zero. Here we give a positive lower bound for this probability depending on x_ that becomes large as 1/2^+. The main inputs in our proofs are a maximal inequality based in relatively high moments for these partial sums combined with a Hal\'asz-Bonami's moment inequality, and also explicit estimates for the partial sums of non-negative multiplicative functions.