Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T08:36:54.411Z Has data issue: false hasContentIssue false
  • English
  • Français

RENEWAL SEQUENCES WITH PERIODIC DYNAMICS

Published online by Cambridge University Press:  25 November 2011

Brian Fralix ,
James Livsey  and
Robert Lund
Brian Fralix
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975. E-mail: [email protected]; [email protected]; [email protected]
James Livsey
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975. E-mail: [email protected]; [email protected]; [email protected]
Robert Lund
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975. E-mail: [email protected]; [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Discrete-time renewal sequences play a fundamental role in the theory of stochastic processes. This article considers periodic versions of such processes; specifically, the length of an interrenewal is allowed to probabilistically depend on the season at which it began. Using only elementary renewal and Markov chain techniques, computational and limiting aspects of periodic renewal sequences are investigated. We use these results to construct a time series model for a periodically stationary sequence of integer counts.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 1 , January 2012 , pp. 1 - 15
Copyright
Copyright © Cambridge University Press 2011

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

1.Asmussen, S. & Thorisson, H. (1987). A Markov chain approach to periodic queues. Journal of Applied Probability 24: 215–225.CrossRefGoogle Scholar
2.Billingsley, P. (1995). Probability and measure, 3rd ed.New York: Wiley.Google Scholar
3.Çinlar, E. (1974). Periodicity in Markov renewal theory. Advances in Applied Probability 6: 61–78.CrossRefGoogle Scholar
4.Cui, Y. & Lund, R. (2009). A new look at time series of counts. Biometrika 96: 781–792.CrossRefGoogle Scholar
5.Feller, W. (1968). An Introduction to probability theory and its applications, Volume I, 3rd ed.New York: Wiley.Google Scholar
6.Gardner, W.A., Napolitano, A., & Paura, L. (2006). Cyclostationarity: Half a century of research. Signal Processing 86: 639–697.CrossRefGoogle Scholar
7.Harrison, J.M. & Lemoine, A.J. (1977). Limit theorems for queues with periodic input. Journal of Applied Probability 14: 566–576.Google Scholar
8.Heyman, D.P. & Whitt, W. (1984). The asymptotic behavior of queues with time-varying arrival rates. Journal of Applied Probability 21: 143–156.CrossRefGoogle Scholar
9.Hurd, H.L. & Miamee, A. (2007). Periodically correlated random sequences: Spectral theory and practice. New York: Wiley.CrossRefGoogle Scholar
10.Jagers, P. & Nerman, O. (1985). Branching processes in a periodically varying environment. Annals of Probability 13: 254–268.CrossRefGoogle Scholar
11.Karlin, S. & Taylor, H.M. (1975). A first course in stochastic processes. New York: Academic Press.Google Scholar
12.Lemoine, A.J (1981). On queues with periodic Poisson input. Journal of Applied Probability 18: 889–900.CrossRefGoogle Scholar
13.Lund, R. (1994). A dam with seasonal input. Journal of Applied Probability 31: 526–541.CrossRefGoogle Scholar
14.Lund, R. & Basawa, I.V. (1999). Modeling for periodically correlated time series, In Asymptotics, Nonparametrics, and Time Series. New York: Dekker, pp. 37–62.Google Scholar
15.Phatarfod, R.M. (1980). The bottomless dam with seasonal inputs. Australian and New Zealand Journal of Statistics 22: 212–217.Google Scholar
16.Resnick, S.I. (1992). Adventures in stochastic processes. Boston: Birkhäuser.Google Scholar
17.Ross, S.M. (1996). Stochastic processes, 2nd ed.New York: Wiley.Google Scholar
18.Smith, W.L. (1958). Renewal theory and its ramifications. Journal of the Royal Statistical Society, Series B 20: 243–302.Google Scholar
19.Thorisson, H. (1985). Periodic regeneration. Stochastic Processes and their Applications 20: 85–104.Google Scholar