Martingale central limit theorem for random multiplicative functions

TL;DR

This paper proves a generalized central limit theorem for sums involving random multiplicative functions, showing the limiting distribution is non-Gaussian with a random variance for a broad class of functions.

Contribution

It establishes a non-Gaussian limit theorem for sums of multiplicative functions multiplied by random multiplicative functions, extending previous results to a wider class of functions.

Findings

The sum normalized to mean square 1 has a non-Gaussian limit distribution.

The result applies to divisor functions with parameter z between 0 and 1/√2.

Includes multiplicative indicator functions with specific prime density properties.

Abstract

Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}\alpha(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian limiting distribution. More precisely, we establish a generalised central limit theorem with random variance determined by the total mass of a random measure associated with $\alpha f$. Our result applies to $d_z$, the $z$-th divisor function, as long as $z$ is strictly between $0$ and $\tfrac{1}{\sqrt{2}}$. Other examples of admissible $f$-s include any multiplicative indicator function with the property that $f(p)=1$ holds for a set of primes of density strictly between $0$ and $\tfrac{1}{2}$.