Functional limit theorems for discounted exponential functional of random walk and discounted convergent perpetuity

Let (ξ1,η1), (ξ2,η2),… be independent and identically distributed R2-valued random vectors. Put S0≔0 and Sk≔ξ1+…+ξk for k∈N. We prove a functional central limit theorem for a discounted exponential functional of the random walk ∑k≥0e−Sk∕t, properly normalized and centered, as t→∞. In combination with a theorem obtained recently in Iksanov et al. (2021) this leads to an ultimate functional central limit theorem for a discounted convergent perpetuity ∑k≥0e−Sk∕tηk+1, again properly normalized and centered, as t→∞. The latter result complements Vervaat’s (1979) one-dimensional central limit theorem. Our argument is different from that used by Vervaat. The functional limit theorem is not informative in the case where ξk=ηk. As a remedy, we show that ∑k≥0e−Sk∕tξk+1 concentrates tightly around the point t in a deterministic manner.