Asymptotic bounds on renewal process stopping times

TL;DR

This paper derives asymptotic bounds for the expected stopping times of renewal processes involving transformed uniform random variables, generalizing previous results for the identity function.

Contribution

It extends known results on renewal process stopping times to a broader class involving increasing bijections, providing a general asymptotic formula.

Findings

Derived the asymptotic expression for expected stopping times $N_f(t)$

Introduced the constant $c_f$ to characterize the bounds

Generalized previous results for the case $f(x)=x$

Abstract

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 $f(x) = x$.