Limit theorems for discounted convergent perpetuities II

Abstract

Let $({\mathrm{\xi }}_1,{\mathrm{\eta }}_1)$, $({\mathrm{\xi }}_2,{\mathrm{\eta }}_2),\dots$ be independent identically distributed ${\mathbb{R}}^2$-valued random vectors. Assuming that ${\mathrm{\xi }}_1$ has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of ${\mathrm{\eta }}_1$ we prove three functional limit theorems for the logarithm of convergent discounted perpetuities ${\sum }_{k\ge 0}{\mathrm{e}}^{{\mathrm{\xi }}_1+\dots +{\mathrm{\xi }}_k-ak}{\mathrm{\eta }}_{k+1}$ 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.