  Suppose that i.i.d. random variables $X_{1}, X_{2}, \ldots$ are chosen
uniformly from $[0,1]$, and let $f: [0,1] \rightarrow [0,1]$ be an increasing
bijection. Define $\mu_{f}$ to be the expected value of $f(X_{i})$ for each
$i$. Define the random variable $K_{f}$ be to be minimal so that $\sum_{i =
1}^{K_{f}} f(X_{i}) > t$ and let $N_{f}(t)$ be the expected value of $K_{f}$.
We prove that if $c_{f} = \frac{\int_{0}^{1} \int_{f^{-1}(u)}^{1} (f(x)-u) dx
du}{\mu_{f}}$, then $N_{f}(t) = \frac{t+c_{f}}{\mu_{f}}+o(1)$. This generalizes
a result of \'{C}urgus and Jewett (2007) on the case $f(x) = x$.
