Let $({\xi }_1,{\eta }_1)$, $({\xi }_2,{\eta }_2),\dots$ be independent and identically distributed ${\mathbb{R}}^2$-valued random vectors. Put $S_0≔0$ and $S_k≔{\xi }_1+\dots +{\xi }_k$ for $k\in \mathbb{N}$. We prove a functional central limit theorem for a discounted exponential functional of the random walk ${\sum }_{k\ge 0}{e}^{-S_k∕t}$, properly normalized and centered, as $t\to \infty$. 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 ${\sum }_{k\ge 0}{e}^{-S_k∕t}{\eta }_{k+1}$, again properly normalized and centered, as $t\to \infty$. 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 ${\xi }_k={\eta }_k$. As a remedy, we show that ${\sum }_{k\ge 0}{e}^{-S_k∕t}{\xi }_{k+1}$ concentrates tightly around the point $t$ in a deterministic manner.

让 $（{\xi }_{\mathrm{1个}}，{\eta }_{\mathrm{1个}}）$， $（{\xi }_{\mathrm{2个}}，{\eta }_{\mathrm{2个}}），\dots$ 独立且分布均匀 ${\mathbb{[R}}^{\mathrm{2个}}$值的随机向量。放${\mathrm{小号}}_0≔0$ 和 ${\mathrm{小号}}_{ķ}≔{\xi }_{\mathrm{1个}}+\dots +{\xi }_{ķ}$ 为了 $ķ\in \mathbb{ñ}$。我们证明了随机游走的折现指数函数的泛函中心极限定理${\sum }_{ķ\ge 0}{Ë}^{-{\mathrm{小号}}_{ķ}∕Ť}$，正确归一化并居中，如 $Ť\to \infty$。结合最近在Iksanov等人获得的一个定理。（2021）这导致了折衷的收敛永久性的终极功能中心极限定理${\sum }_{ķ\ge 0}{Ë}^{-{\mathrm{小号}}_{ķ}∕Ť}{\eta }_{ķ+\mathrm{1个}}$，再次正确归一化和居中，如 $Ť\to \infty$。后者的结果是对Vervaat（1979）的一维中心极限定理的补充。我们的论点与Vervaat所用的论点不同。在以下情况下，函数极限定理不提供任何信息${\xi }_{ķ}={\eta }_{ķ}$。作为补救措施，我们表明${\sum }_{ķ\ge 0}{Ë}^{-{\mathrm{小号}}_{ķ}∕Ť}{\xi }_{ķ+\mathrm{1个}}$ 紧紧地围绕着重点 $Ť$ 以确定的方式。