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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.
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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 

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Footnotes

Research supported by NSF Grant NCR 92-11343.

References

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