A few notes on the asymptotic behavior of Rademacher random multiplicative functions

Abstract

Let $X_p, p\in\cP$ be a sequence of independent random variables s.t. $\bbP(X_p=\pm 1)=1/2$. Let $\te_j=\prod_{p|j}X_p$ if $j$ is square free and $\te_j=0$ otherwise. Denote $S_n=\sum_{\ell=1}^n\te_\ell$. The from this point of view proving limit theorems for $S_n$ is natural problem, since $S_n$ mimics the behavior of $e^{\sqrt{\ln(\beta)}}$. It is a natural guiding conjecture that $S_n/\sqrt n$ obeys the central limit theorem (CLT). However, S. Chatterjee conjectured (as expressed in \cite{[25]}) that the CLT should not hold. Chatterjee's conjecture was proved by Harper \cite{[17]}, and by now it is a direct consequence of a more recent breakthrough by Harper \cite{Har20} that $\frac{S_n}{b_n}\to 0$ in $L^1$, where $b_n=(n^{1/2}(\ln(\ln(n)))^{-1/4})u_n, u_n\to\infty$. In particular $S_n/\sqrt n\to 0$. Nevertheless, the question whether there exists a sequence $a_n=o(b_n)$ such that $S_n/a_n$ converges to some limit remains a mystery. Note that the corresponding problem in the Steinhaus Setting was recently resolved by \cite{Gor1}. In this paper make an attempt to shed some light on the convergence of $S_n/a_n$. Additionally, we obtain explicit estimates on hight moments of $S_n$ without restrictions on the size of the moment compared to $n$ like in \cite[Theorem 1.2]{Har19}, which is of independent interest. This is achieved by a martingale argument together with the Burkholder inequality, and it has applications in a natural number theoretic combinatorial problem. Using martingale techniques we will also obtain exponential concentration inequalities for $S_n$ (in the large deviations regime)