# On consecutive values of random completely multiplicative functions

## TL;DR

This paper investigates the independence and empirical distribution convergence of consecutive values of random completely multiplicative functions, revealing conditions under which these values behave independently and their empirical measures stabilize almost surely.

## Contribution

It establishes independence of consecutive values for large indices under specific distributions and proves almost sure convergence of empirical measures for these functions.

## Key findings

- Consecutive values become independent for large n under certain distributions.
- Empirical measures of the functions converge almost surely as N increases.
- Rate of convergence of the empirical measure is estimated.

## Abstract

In this article, we study the behavior of consecutive values of random completely multiplicative functions $(X_n)_{n \geq 1}$ whose values are i.i.d. at primes. We prove that for $X_2$ uniform on the unit circle, or uniform on the set of roots of unity of a given order, and for fixed $k \geq 1$, $X_{n+1}, \dots, X_{n+k}$ are independent if $n$ is large enough. Moreover, with the same assumption, we prove the almost sure convergence of the empirical measure $N^{-1} \sum_{n=1}^N \delta_{(X_{n+1}, \dots, X_{n+k})}$ when $N$ goes to infinity, with an estimate of the rate of convergence. At the end of the paper, we also show that for any probability distribution on the unit circle followed by $X_2$, the empirical measure converges almost surely when $k=1$.