Arithmetic properties of multiplicative integer-valued perturbed random walks

TL;DR

This paper investigates the arithmetic structure of multiplicative perturbed random walks, analyzing their prime factorization and least common multiples, and establishing distributional limit theorems for these properties.

Contribution

It introduces a novel analysis of the arithmetic properties of multiplicative perturbed random walks, including prime counts and LCM behavior, with new limit theorems.

Findings

Distributional limit theorems for prime counts

Asymptotic behavior of least common multiples

Characterization of prime factorization patterns

Abstract

Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent identically distributed $\mathbb{N}^2$-valued random vectors with arbitrarily dependent components. The sequence $(\Theta_k)_{k\in\mathbb{N}}$ defined by $\Theta_k=\Pi_{k-1}\cdot\eta_k$, where $\Pi_0=1$ and $\Pi_k=\xi_1\cdot\ldots\cdot \xi_{k}$ for $k\in\mathbb{N}$, is called a multiplicative perturbed random walk. We study arithmetic properties of the random sets $\{\Pi_1,\Pi_2,\ldots, \Pi_k\}\subset \mathbb{N}$ and $\{\Theta_1,\Theta_2,\ldots, \Theta_k\}\subset \mathbb{N}$, $k\in\mathbb{N}$. In particular, we derive distributional limit theorems for their prime counts and for the least common multiple.