Arithmetic properties of multiplicative integer-valued perturbed random walks

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.