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

*Alexander Iksanov* and
*Oleh Kondratenko*

*Statistics & Probability Letters*, 2021, vol. 176, issue C

**Abstract:**
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.

**Keywords:** Exponential functions of random walk; Functional central limit theorem; Perpetuity; Standard random walk (search for similar items in EconPapers)

**Date:** 2021

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:stapro:v:176:y:2021:i:c:s0167715221001103

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**DOI:** 10.1016/j.spl.2021.109148

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