Limit theorems for discounted convergent perpetuities

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. We prove a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm for the convergent perpetuities ${\sum }_{k\ge 0}{b}^{{\mathrm{\xi }}_1+\dots +{\mathrm{\xi }}_k}{\mathrm{\eta }}_{k+1}$ 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.