Doubly Stochastic Poisson Processes
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529 Jan Grandell
Doubly Stochastic Poisson Processes
Springer-Verlag Berlin. Heidelberg New York 1976
Author Jan Grandell Department of Mathematics The Royal Institute of Technology S-10044 Stockholm 70
Library of Congress Cataloging in Publication Data
Grandell, Jan, 194~iDoubly stochastic Poisson processes. (Lecture notes in mathematics ; 529) Bibliography: p. Includes index. 1. Poisson processes, Doubly stochastic. 2. Measure theory. 3. Prediction theory. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 529. QA3.L28 vol. 529 [QA274.42] 510'.8s [519.2'3]
76-20626
A M S Subject Classifications (1970): 60F05, 6 0 G 2 5 , 6 0 G 5 5 , 62M15 ISBN 3-540-0??95-2 ISBN 0 - 3 8 ? - 0 ? ? 9 5 - 2
Springer-Verlag Berlin 9Heidelberg 9N e w York Springer-Verlag N e w York Heidelberg 9 9Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9by Springer-Verlag Berlin. Heidelberg 1976 Printed in Germany. Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
PREFACE
The doubly stochastic Poisson process is a generalization of the ordinary Poisson process in the sense that stochastic variation in the intensity is allowed. Some authors call these processes processes'
'Cox
since they were proposed by Cox (1955). Later on Mecke
(1968) studied doubly stochastic Poisson processes within the framework of the general theory of point processes and random measures.
Point processes have been studied from both a theoretical and a practical point of view. Good expositions of theoretical aspects are given by Daley and Vere-Jones
(1972), Jagers (1974), Kallenberg
(1975:2) and Kerstan~ Matthes and Mecke
(1974). Accounts of more
practical aspects are given by Cox and Lewis (1966) and Snyder (1975).
The exposition in this monograph is based on the general theory of point processes and random measures, but much of it can be read without knowledge of that theory. My objective is to place myself somewhere between the purely theoretical school and the more applied one, since doubly stochastic Poisson processes are of both theoretical and practical interest.
I am quite aware of the risk that some readers
will find this monograph rather shallow while others will find it too abstract. Of course I hope - although perhaps in vain - that a reader who is from the beginning only interested in applications will also find some of the more theoretical parts worth reading. I have, however, tried to make most of the more applied parts understandable without knowledge of the more abstract parts. Also in most of the more theoretical parts I have included examples and numerical illustrations.
JV
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