Asymptotic bounds on renewal process stopping times

Abstract

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 ${\sum }_{i=1}^{K_f}f(X_i)>t$ and let $N_f(t)$ be the expected value of $K_f$. We prove that if $c_f=\frac{{\int }_0^1{\int }_{f^{-1}(u)}^1(f(x)-u)dxdu}{{\mu }_f}$, then $N_f(t)=\frac{t+c_f}{{\mu }_f}+o(1)$. This generalizes a result of Ćurgus and Jewett (2007) on the case $f(x)=x$.

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