## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60K05: Renewal theory

**Keywords**

Renewal function bound Fourier methods

#### Citation

Daley, D. J. Upper Bounds for the Renewal Function Via Fourier Methods. Ann. Probab. 6 (1978), no. 5, 876--884. doi:10.1214/aop/1176995434. https://projecteuclid.org/euclid.aop/1176995434