Sign changes of the partial sums of a random multiplicative function III: Average

TL;DR

This paper establishes a lower bound on the average number of sign changes in partial sums of a Rademacher random multiplicative function, extending classical results to broader dependent systems using advanced moment analysis.

Contribution

It introduces a new method for counting sign changes in dependent orthogonal systems, extending Erd ext{"o}s and Hunt's classical results to larger classes of dependencies.

Findings

Average sign changes grow at least as fast as (log x)(log log x)^{-1/2-ε}.

Method applies to systems with certain dependency structures beyond i.i.d. variables.

Utilizes Harper's improved small moment estimates for partial sums.

Abstract

Let $V(x)$ be the number of sign changes of the partial sums up to $x$, say $M_f(x)$, of a Rademacher random multiplicative function $f$. We prove that the averaged value of $V(x)$ is at least $\gg (\log x)(\log\log x)^{-1/2-\epsilon}$. Our new method applies for the counting of sign changes of the partial sums of a system of orthogonal random variables having variance $1$ under additional hypothesis on the moments of these partial sums. In particular, we extend to larger classes of dependencies an old result of Erd\H{o}s and Hunt on sign changes of partial sums of i.i.d. random variables. In the arithmetic case, the main input in our method is the ``\textit{linearity}'' phase in $1\leq q\leq 1.9$ of the quantity $\log \mathbb{E} |M_f(x)|^q$, provided by the Harper's \textit{better than squareroot cancellation} phenomenon for small moments of $M_f(x)$.