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A martingale approach to the PASTA property

Part of: Limit theorems Special processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Michael Scheutzow
Michael Scheutzow*
Affiliation:
Technische Universität Berlin
*
∗ Postal address: Fachbereich Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-1000 Berlin 12, Germany.
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Abstract

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

Keywords

LAW OF LARGE NUMBERS FOR MARTINGALESSTOCHASTIC INTEGRALPOISSON PROCESSQUEUEING THEORYRANDOM MEASURE

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60F15: Strong theorems 60H05: Stochastic integrals 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 252 - 257
Copyright
Copyright © Applied Probability Trust 1993 

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