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Universitext
Achim Klenke
Probability Theory
A Comprehensive Course
123
Prof. Dr. Achim Klenke
Institut f
¨
ur Mathematik
Johannes Gutenberg-Universit
¨
at Mainz
Staudingerweg
9
55099
Mainz
Germany
ISBN: 978-1-84800-047-6 e-ISBN: 978-1-84800-048-3
DOI: 10.1007/978-1-84800-048-3
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2007939558
Mathematics Subject Classification (2000): 60-01; 28-01
Translation from the German language edition:
Wahrscheinlichkeitstheorie by Achim Klenke
Copyright
c
Springer Verlag Berlin Heidelberg 2006
Springer is a part of Springer Science+Business Media
All Rights Reserved
c
Springer-Verlag London Limited 2008
Apart from any fair dealing for the purposes of research or private study, or criticism or review,
as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be
reproduced, stored or transmitted, in any form or by any means, with the prior permission in
writing of the publishers, or in the case of reprographic reproduction in accordance with the
terms of licenses issued by the Copyright Licensing Agencies. Enquiries concerning repro-
duction outside those terms should be sent to the publishers.
The use of registered name, trademarks, etc., in this publication does not imply, even in the
absence of a specific statement, that such names are exempt from the relevant laws and regu-
lations and therefore free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the
information contained in this book and cannot accept any legal responsibility or liability for
any errors or omissions that may be made.
Printed on acid-free paper
987654321
Springer Science+Business Media
springer.com
Preface
This book is based on two four-hour courses on advanced probability theory that I
have held in recent years at the universities of Cologne and Mainz. It is implicitly as-
sumed that the reader has a certain familiarity with the basic concepts of probability
theory, although the formal framework will be fully developed in this book.
The aim of this book is to present the central objects and concepts of probability
theory: random variables, independence, laws of large numbers and central limit
theorems, martingales, exchangeability and infinite divisibility, Markov chains and
Markov processes, as well as their connection with discrete potential theory, cou-
pling, ergodic theory, Brownian motion and the It
ˆ
o integral (including stochastic
differential equations), the Poisson point process, percolation and the theory of large
deviations.
Measure theory and integration are necessary prerequisites for a systematic proba-
bility theory. We develop it only to the point to which it is needed for our purposes:
construction of measures and integrals, the Radon-Nikodym theorem and regular
conditional distributions, convergence theorems for functions (Lebesgue) and mea-
sures (Prohorov) and construction of measures in product spaces. The chapters on
measure theory do not come as a block at the beginning (although they are written
such that this would be possible; that is, independent of the probabilistic chapters)
but are rather interlaced with probabilistic chapters that are designed to display the
power of the abstract concepts in the more intuitive world of probability theory. For
example, we study percolation theory at the point where we barely have measures,
random variables and independence; not even the integral is needed. As the only ex-
ception, the systematic construction of independent random variables is deferred to
Chapter 14. Although it is rather a matter of taste, I hope that this setup helps to
motivate the reader throughout the measure-theoretical chapters.
Those readers with a solid measure-theoretical education can skip in particular the
first and fourth chapters and might wish only to look up this or that.
VI Preface
In the first eight chapters, we lay the foundations that will be needed in all subsequent
chapters. After that, there are seven more or less independent parts, consisting of
Chapters 9–12, 13, 14, 15–16, 17–19, 20 and 23. The chapter on Brownian motion
(21) makes reference to Chapters 9–15. Again, after that, the three blocks consisting
of Chapters 22, 24 and 25–26 can be read independently.
I should like to thank all those who read the manuscript and the German original
version of this book and gave numerous hints for improvements: Roland Alkemper,
Ren
´
e Billing, Dirk Br
¨
uggemann, Anne Eisenb
¨
urger, Patrick Jahn, Arnulf Jentzen,
Ortwin Lorenz, L. Mayer, Mario Oeler, Marcus Sch
¨
olpen, my colleagues Ehrhard
Behrends, Wolfgang B
¨
uhler, Nina Gantert, Rudolf Gr
¨
ubel, Wolfgang K
¨
onig, Peter
M
¨
orters and Ralph Neininger, and in particular my colleague from Munich Hans-
Otto Georgii. Dr John Preater did a great job language editing the English manuscript
and also pointing out numerous mathematical flaws.
I am especially indebted to my wife Katrin for proofreading the English manuscript
and for her patience and support.
I would be grateful for further suggestions, errors etc. to be sent by e-mail to
math@aklenke.de
Mainz, Achim Klenke
October 2007
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