Arithmetic properties of multiplicative integer-valued perturbed random walks

Abstract

Let $({\xi }_1,{\eta }_1)$, $({\xi }_2,{\eta }_2),\dots$ 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 ${\Pi }_k={\xi }_1\cdot \dots \cdot {\xi }_k$ for $k\in \mathbb{N}$, is called a multiplicative perturbed random walk. We study arithmetic properties of the random sets $\{{\Pi }_1,{\Pi }_2,\dots ,{\Pi }_k\}\subset \mathbb{N}$ and $\{{\Theta }_1,{\Theta }_2,\dots ,{\Theta }_k\}\subset \mathbb{N}$, $k\in \mathbb{N}$. In particular, we derive distributional limit theorems for their prime counts and for the least common multiple.

16 pages