Limit theorems for discounted convergent perpetuities II

Abstract

Let $({\xi }_1,{\eta }_1)$, $({\xi }_2,{\eta }_2),\dots$ 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 ${\sum }_{k\ge 0}{e}^{{\xi }_1+\dots +{\xi }_k-ak}{\eta }_{k+1}$ 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.

23 pages