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A martingale view of Blackwell’s renewal theorem and its extensions to a general counting process

Part of: Special processes Markov processes

Published online by Cambridge University Press:  30 July 2019

Daryl J. Daley  and
Masakiyo Miyazawa
Daryl J. Daley*
Affiliation:
The University of Melbourne
Masakiyo Miyazawa*
Affiliation:
Tokyo University of Science and Chinese University of Hong Kong, Shenzhen
*
*Postal address: Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia.
**Postal address: Department of Information Sciences, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan.
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Abstract

Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell’s renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell’s renewal theorem holds.

Keywords

Renewal processsemimartingaleBlackwell’s renewal theoremkey renewal theoremWald’s identitystationary intervalsPalm distributionmodulated renewal processSmith’s asymptotic formulae

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60K37: Processes in random environments
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 602 - 623
Copyright
© Applied Probability Trust 2019 

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