Asymptotic bounds on renewal process stopping times

Suppose that i.i.d. random variables $X_1,X_2,\dots$ are chosen uniformly from $[0,1]$, and let $f:[0,1]\to [0,1]$ 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 $\sum_{i = 1}^{K_{f}} f(X_{i}) > t$andlet$N_{f}(t)$betheexpectedvalueof$K_{f}$. 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 $f(x)=x$.