Limit theorems for discounted convergent perpetuities II

TL;DR

This paper establishes three functional limit theorems and a law of the iterated logarithm for the logarithm of convergent discounted perpetuities under various distributional assumptions, extending previous research in the area.

Contribution

It introduces new limit theorems for discounted perpetuities with specific distributional assumptions, advancing understanding of their asymptotic behavior.

Findings

Proved three functional limit theorems for the logarithm of discounted perpetuities.

Established a law of the iterated logarithm related to these perpetuities.

Extended previous results to new distributional settings.

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.