Limit theorems for discounted convergent perpetuities

Abstract

Let (ξ1, η1), (ξ2, η2), . . . be independent identically distributed 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 k≥0 b ξ 1 +...+ξ k η k+1 as b → 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.

Limit theorems for discounted convergent perpetuities II

“…One option is 1) or equivalently k≥0 e −S k /t η k+1 for t > 0. In the recent article [13] basic limit theorems for X(b), properly normalized, as b → 1−, were proved, namely, a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm. To be more specific, we state a combination of Theorem 1.2 and one part of Theorem 1.5 in [13] as Proposition 1.1.…”

“…In the recent article [13] basic limit theorems for X(b), properly normalized, as b → 1−, were proved, namely, a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm. To be more specific, we state a combination of Theorem 1.2 and one part of Theorem 1.5 in [13] as Proposition 1.1. Denote by C = C(0, ∞) the space of continuous functions defined on (0, ∞) equipped with the locally uniform topology.…”