Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T12:42:43.300Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alsmeyer, G. (1997). The Markov renewal theorem and related results. Markov Process. Related Fields 3, 103–127.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, second edition (Applications of Mathematics: Stochastic Modelling and Applied Probability 51). Springer, New York.Google Scholar
Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, second edition (Applications of Mathematics 26). Springer, Berlin.Google Scholar
Bremaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer.Google Scholar
Chen, H. and Yao, D. (2001). Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Springer, New York.Google Scholar
Cinlar, E. (1975). Introduction to Stochastic Processes. Prentice Hall, New Jersey.Google Scholar
Daley, D. (2017). Renewal function asymptotics refined à la Feller. Probab. Math. Statist. 37, 291–298.Google Scholar
Daley, D. and Mohan, N. (1978). Asymptotic behaviour of the variance of renewal processes and random walks. Ann. Prob. 6, 516–521.CrossRefGoogle Scholar
Daley, D. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes (Springer Series in Statistics). Wiley, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, vol. II, second edition. Wiley, New York.Google Scholar
Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions (Springer Series in Operations Research and Financial Engineering). Springer.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, second edition. Springer, Berlin.Google Scholar
Kallenberg, O. (2001). Foundations of Modern Probability, second edition (Springer Series in Statistics, Probability and its Applications). Springer, New York.Google Scholar
Miyazawa, M. (1977). Time and customer processes in queues with stationary inputs. J. Appl. Prob. 14, 349–357.CrossRefGoogle Scholar
Miyazawa, M. (2017). Martingale approach for tail asymptotic problems in the generalized Jackson network. Probab. Math. Statist. 37, 395–430.Google Scholar
Miyazawa, M. (2017). A unified approach for large queue asymptotics in a heterogeneous multiserver queue. Adv. Appl. Prob. 49, 182–220.CrossRefGoogle Scholar
Pang, G., Talreja, R. and Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4, 193–267.CrossRefGoogle Scholar
Sgibnev, M. (1981). On the renewal theorem in the case of infinite variance. Siberian Math. J. 22, 787–796.CrossRefGoogle Scholar
Smith, W. L. (1959). On the cumulants of renewal processes. Biometrika 46, 1–29.CrossRefGoogle Scholar
Van Der Vaart, A. W. (2006). Martingales, diffusions and financial mathematics. Lecture notes, available at http://www.math.vu.nl/sto/.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.CrossRefGoogle Scholar
Whitt, W. (2007). Proofs of the martingale FCLT. Probab. Surv. 4, 268–302.CrossRefGoogle Scholar