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# Probability Theory A Comprehensive Course

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Probability Theory A Comprehensive Course

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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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