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October, 1978 Upper Bounds for the Renewal Function Via Fourier Methods
D. J. Daley
Ann. Probab. 6(5): 876-884 (October, 1978). DOI: 10.1214/aop/1176995434


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 $C = 1$.


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D. J. Daley. "Upper Bounds for the Renewal Function Via Fourier Methods." Ann. Probab. 6 (5) 876 - 884, October, 1978.


Published: October, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0388.60087
MathSciNet: MR494547
Digital Object Identifier: 10.1214/aop/1176995434

Primary: 60K05

Keywords: Fourier methods , Renewal function bound

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 5 • October, 1978
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