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On the weak convergence of stochastic processes with embedded point processes

Published online by Cambridge University Press:  01 July 2016

Panagiotis Konstantopoulos  and
Jean Walrand
Panagiotis Konstantopoulos*
Affiliation:
University Of California, Berkeley
Jean Walrand*
Affiliation:
University Of California, Berkeley
*
∗Postal address: Department of Electrical Engineering and Computer Sciences and Electronics Research Laboratory, University of California, Berkeley, CA 94720, USA.
∗Postal address: Department of Electrical Engineering and Computer Sciences and Electronics Research Laboratory, University of California, Berkeley, CA 94720, USA.
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Abstract

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We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

Keywords

STATIONARY POINT PROCESSESPALM DISTRIBUTIONSMIXING
Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 473 - 475
Copyright
Copyright © Applied Probability Trust 1988 

References

Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queues. Lecture Notes in Statistics 41, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Akademie Verlag, Berlin.Google Scholar
König, D. and Schmidt, V. (1986) Limit theorems for single-server feedback queues controlled by a general class of marked point processes. Theory Prob. Appl. 30, 712–719.Google Scholar
Lindvall, T. (1982) On coupling of continuous-time renewal processes. J. Appl. Prob. 19, 82–89.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, New York.Google Scholar
Miyazawa, M. (1977) Time and customer processes in queues with stationary inputs. J. Appl. Prob. 14, 349–357.CrossRefGoogle Scholar
Neveu, J. (1976) Sur les mesures de Palm de deux processus ponctuelles stationnaires. Z. Wahrscheinlichkeitsth. 34, 199–203.Google Scholar
Neveu, J. (1977) Processus ponctuelles. In Lecture Notes in Mathematics 598, Springer-Verlag, Berlin, 249–447.Google Scholar