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2020 The remainder in the renewal theorem
Ron Doney
Electron. Commun. Probab. 25: 1-8 (2020). DOI: 10.1214/20-ECP287

Abstract

If the step distribution in a renewal process has finite mean and regularly varying tail with index $-\alpha $, $1<\alpha <2$, the first two terms in the asymptotic expansion of the renewal function have been known for many years. Here we show that, without making any additional assumptions, it is possible to give, in all cases except for $\alpha =3/2$, the exact asymptotic behaviour of the next term. In the case $\alpha =3/2$ the result is exact to within a slowly varying correction. Similar results are shown to hold in the random walk case.

Citation

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Ron Doney. "The remainder in the renewal theorem." Electron. Commun. Probab. 25 1 - 8, 2020. https://doi.org/10.1214/20-ECP287

Information

Received: 25 September 2019; Accepted: 9 January 2020; Published: 2020
First available in Project Euclid: 28 January 2020

zbMATH: 1434.60251
MathSciNet: MR4066298
Digital Object Identifier: 10.1214/20-ECP287

Subjects:
Primary: 60G50 , 60K05

Keywords: asymtotic stability , Random walks , regular variation , Renewal theorem

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