Asymptotic bounds on renewal process stopping times

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 $μ_{f}$ 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 $c_{f} = \frac{\int_{0}^{1} \int_{f^{-1}(u)}^{1} (f(x)-u) dx du}{μ_{f}}$, then $N_{f}(t) = \frac{t+c_{f}}{μ_{f}}+o(1)$. This generalizes a result of Ćurgus and Jewett (2007) on the case $f(x) = x$.

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