Abstract
The general renewal equation and real variable methods are used to show that for a renewal process with generic lifetime random variable $X \geqq 0$ having distribution $F$ and finite first and second moments $EX = \lambda^{-1}$ and $EX^2$, the renewal function $U(x) = \sum^\infty_0 F^{n^\ast(x)$ satisfies $U(x) \leqq \lambda x_+ + C\lambda^2EX^2$ for a certain constant $C$ independent of $F$. Stone (1972) showed that $1 \leqq C \leqq 2.847 \cdots$; it is proved here that $C \leqq 1.3186 \cdots$ and conjectured that $C = 1$.
Citation
D. J. Daley. "Another Upper Bound for the Renewal Function." Ann. Probab. 4 (1) 109 - 114, February, 1976. https://doi.org/10.1214/aop/1176996188
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