Limit theorems for discounted convergent perpetuities II

Abstract

Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent identically distributed $\mathbb{R}^2$-valued random vectors. Assuming that $\xi_1$ has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of $\eta_1$ we prove three functional limit theorems for the logarithm of convergent discounted perpetuities $\sum_{k\geq 0}e^{\xi_1+\ldots+\xi_k-ak}\eta_{k+1}$ as $a\to 0+$. Also, we prove a law of the iterated logarithm which corresponds to one of the aforementioned functional limit theorems. The present paper continues a line of research initiated in the paper Iksanov, Nikitin and Samoillenko (2022), which focused on limit theorems for a different type of convergent discounted perpetuities.