Limit theorems for discounted convergent perpetuities

Abstract

Let $({\xi }_1,{\eta }_1)$, $({\xi }_2,{\eta }_2),\dots$ be independent identically distributed ${\mathbb{R}}^2$-valued random vectors. We prove a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm for convergent perpetuities ${\sum }_{k\ge 0}{b}^{{\xi }_1+\dots +{\xi }_k}{\eta }_{k+1}$ as $b\to 1-$. Under the standard actuarial interpretation, these results correspond to the situation when the actuarial market is close to the customer-friendly scenario of no risk.

27 pages, submitted to a journal