## The Annals of Probability

### Upper Bounds for the Renewal Function Via Fourier Methods

D. J. Daley

#### Abstract

Stone has used Fourier analytic methods to show that the renewal function $U(x) = \sum^\infty_0 F^{n\ast}(x)$ for a random variable $X$ with distribution function $F$, finite second moment and positive mean $\lambda^{-1} = EX$, is bounded above by $\lambda x_+ + C\lambda^2EX^2$ for a universal constant $C, 1 \leqq C < 3$. This paper refines his method to prove that $C < 2.081$, and shows that within certain constraints the smallest upper bound on $C$ that the method will yield is 1.809. Various authors' work on the simpler case where $X \geqq 0$ is summarized: the best result is the earliest published one, due to Lorden, who showed that then $C = 1$.

#### Article information

Source
Ann. Probab., Volume 6, Number 5 (1978), 876-884.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995434

Digital Object Identifier
doi:10.1214/aop/1176995434

Mathematical Reviews number (MathSciNet)
MR494547

Zentralblatt MATH identifier
0388.60087

JSTOR