Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-18T08:03:28.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Functional approximation theorems for controlled renewal processes

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Takis Konstantopoulos ,
Spyros N. Papadakis  and
Jean Walrand
Takis Konstantopoulos*
Affiliation:
University of Texas at Austin
Spyros N. Papadakis*
Affiliation:
University of California, Berkeley
Jean Walrand*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Electrical and Computer Engineering, University of Texas, Austin, TX 78712, USA.
∗∗ Postal address: Department of EECS, University of California, Berkeley, CA 94720, USA.
∗∗ Postal address: Department of EECS, University of California, Berkeley, CA 94720, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.

Keywords

FUNCTIONAL LAW OF LARGE NUMBERSFUNCTIONAL CENTRAL LIMIT THEOREMPOINT PROCESSESMARTINGALES

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 765 - 776
Copyright
Copyright © Applied Probability Trust 1994 

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.)

Footnotes

Research supported by NSF Grant NCR 92-11343.

References

[1] Anantharam, V. and Konstantopoulos, T. (1992) A functional central limit theorem for the jump processes of a uniform Markov process. Preprint, ECE Dept., U.T. Austin.Google Scholar
[2] Anantharam, V. and Konstantopoulos, T. (1992) Same title as above, Proc. 26th Annual Conf. Info. Sci. and Systems, Princeton University.Google Scholar
[3] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[4] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[5] Brémaud, P. (1981) Point Processes and Queues, Martingale Dynamics. Springer-Verlag, New York.Google Scholar
[6] Dacunha-Castelle, D. and Duflo, M. (1986) Probability and Statistics, Vol. 2. Springer-Verlag, New York.Google Scholar
[7] Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.CrossRefGoogle Scholar
[8] Jacod, J. and Shiryayev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[9] Kogan, Ya. A., Liptser, R. Sh. and Smorodinskii, A. V. (1986) Gaussian diffusion approximations of closed Markov models of computer networks. Probl. Pered. Inform 22, 49–65 (English translation).Google Scholar
[10] Liptser, R. Sh. and Shiryayev, A. N. (1978) Statistics of Random Processes II. Springer-Verlag, Berlin.Google Scholar
[11] Liptser, R. Sh. and Shiryayev, A. N. (1980). A functional central limit theorem for semimartingales. Theory Prob. Appl. 25, 667–688.Google Scholar
[12] Liptser, R. Sh. and Shiryayev, A. N. (1989) Theory of Martingales. Kluwer, Boston.Google Scholar
[13] Papadakis, S. N. (1993) On the Macroscopic Behavior of a Class of Controlled Queueing Networks. M.Sc. Report, Dept, of EECS, University of California, Berkeley.Google Scholar
[14] Papadakis, S. N. (1993) A Functional Central Limit Theorem for Controlled Renewal Processes. , Dept. of Mathematics, University of California, Berkeley.Google Scholar