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Event and time averages: a review

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Pierre Brémaud ,
Raghavan Kannurpatti  and
Ravi Mazumdar
Pierre Brémaud*
Affiliation:
CNRS and École Polytechnique
Raghavan Kannurpatti*
Affiliation:
Columbia University
Ravi Mazumdar*
Affiliation:
Université du Québec
*
∗ Postal address: Laboratoire des Signaux et Systèmes, CNRS-ESE, Plateau du Moulon, 91190 Gif-sur-Yvette Cédex, France.
∗∗ Postal address: Department of Electrical Engineering and Center for Telecommunications Research, Columbia University, New York, NY 10027, USA.
∗∗∗ Postal address: INRS-Télécommunications, Université du Québec, Ile-des-Soeurs, PQ, H3E 1H6, Canada.
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Abstract

This article reviews results related to event and time averages (EATA) for point process models, including PASTA, ASTA and ANTIPASTA under general hypotheses. In particular, the results for the stationary case relating the Palm and martingale approach are reviewed. The non-stationary case is discussed in the martingale framework where minimal conditions for ASTA generalizing earlier work are presented in a unified framework for the discrete- and continuous-time cases. In addition, necessary and sufficient conditions for ASTA to hold in the stationary case are discussed in the case even when stochastic intensities may not exist and a short proof of the ANTIPASTA results known to date are given.

Keywords

STATIONARY POINT PROCESSESPALM PROBABILITIESSTOCHASTIC INTENSITYNON-STATIONARY POINT PROCESSESMARTINGALESRADON-NIKODYM DERIVATIVESQUEUEING PROCESSES

MSC classification

Primary: 60G55: Point processes
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 377 - 411
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

This paper was presented at the TIMS/ORSA Conference, Monterey, California, January 1991.

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