Amarts and Set Function Processes
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1042 Allan Gut Klaus D. Schmidt
Amarts and Set Function Processes
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Authors Allan Gut Department of Mathematics, University of Uppsala Thunbergsv~.gen 3, 75238 Uppsala, Sweden Klaus D. Schmidt Seminar f(~r Statistik, Universit~t Mannheim, A 5 6800 Mannheim, Federal Republic of Germany
AMS Subject Classifications (1980): 6 0 G 4 8 ; 6 0 G 4 0 , 6 0 G 4 2 ISBN 3-540-12867-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12867-0 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright.All rights are reserved,whetherthe whole or partof the material is concerned, specificallythose of translation,reprinting, re-useof illustrations,broadcasting, reproduction by photocopying machineor similar means,and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payableto "VerwertungsgesellschaftWort~, Munich. © by Springer-VerlagBerlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 214613140-543210
Ama
r t s
set
F u n c t i o n
Allan An
and
Gut:
introduction
asymptotic
Klaus Amarts
Allan Amarts
P r o c e s s e s
D.
to
martingales
theor~
of
....................
Schmidt:
- a measure
Gut
the
and
Klaus
theoretic
D.
- a bibliography
approach
51
Schmidt: ...................
237
AN INTRODUCTION TO THE THEORY OF
ASYHPTOTICHARTINC~ES
By Allan Gut
Contents
page
Preface
4
Introduction
5
I. History
9
2. Basic properties
14
3. Convergence
23
4. Some examples
31
5. Stability
35
6. The Riesz decomposition
40
7. Two further generalizations of martingales
44
References
46
Preface The material of these notes is based on a series of lectures on real-valued asymptotic martingales
(amarts) held at the Department of
Mathematics at Uppsala University in spring 1979. The purpose of the lectures (and now also of these notes) was (is) to introduce an audience~ familiar to martingale theory~ to the theory of asymptotic martingales. A most important starting point for the development of amart theory was made by Austin, Edgar and Ionescu Tulcea (1974), who presented a beautiful device for proving convergence results. In Edgar, and Sucheston (1976a) the first more systematic treatment of asymptotic martingales was made. Since then several articles on asymptotic martingales have appeared in various journals, l~i~ book therefore ends with a list of references containing all papers related to the theory of asymptotic martingales that wehave been able to trace, whether cited in the text or not.
Introduction We begin by defining asymptotic martingales (amarts) and by briefly investigating how they are related to martingales, submartingales, quasimartingales and other generalizations of martingales. This is then followed by a section on the history of asymptotic martingales after which the more detailed presentation of the theory begins. In this introductory part we consider, for simplicity, on
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