Arithmetic properties of multiplicative integer-valued perturbed random walks

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 $({\mathrm{\Theta }}_k{)}_{k\in \mathbb{N}}$ defined by ${\mathrm{\Theta }}_k={\mathrm{\Pi }}_{k-1}\cdot {\eta }_k$, where ${\mathrm{\Pi }}_0=1$ and ${\mathrm{\Pi }}_k={\xi }_1\cdot \dots \cdot {\xi }_k$ for $k\in \mathbb{N}$, is called a multiplicative perturbed random walk. We study arithmetic properties of the random sets $\{{\mathrm{\Pi }}_1,{\mathrm{\Pi }}_2,\dots ,{\mathrm{\Pi }}_k\}\subset \mathbb{N}$ and $\{{\mathrm{\Theta }}_1,{\mathrm{\Theta }}_2,\dots ,{\mathrm{\Theta }}_k\}\subset \mathbb{N}$, $k\in \mathbb{N}$. In particular, we derive distributional limit theorems for their prime counts and for the least common multiple.