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Limit theorems for discounted convergent perpetuities

Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ 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 $\sum_{k\geq 0}b^{ξ_1+\ldots+ξ_k}η_{k+1}$ 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.