## The Annals of Probability

### Another Upper Bound for the Renewal Function

D. J. Daley

#### 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 $C = 1$.

#### Article information

Source
Ann. Probab., Volume 4, Number 1 (1976), 109-114.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996188

Digital Object Identifier
doi:10.1214/aop/1176996188

Mathematical Reviews number (MathSciNet)
MR391291

Zentralblatt MATH identifier
0329.60055

JSTOR
links.jstor.org

Subjects
Primary: 60K05: Renewal theory

Keywords
Renewal function bound

#### Citation

Daley, D. J. Another Upper Bound for the Renewal Function. Ann. Probab. 4 (1976), no. 1, 109--114. doi:10.1214/aop/1176996188. https://projecteuclid.org/euclid.aop/1176996188