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Probability with Martingales
David Williams
Statistical Laboratory, DPMMS
Cambridge University
Th~
right ofthe
Uni...,,,ity of
Combridg~
to print ond
ull
011
manner of
books
was gronltd
by
Henry
Vlll
in
I5J-I.
The
Uni.~"ity
hos printed
and published continuously
since 158-1.
CAMBRIDGE UNIVERSITY PRESS
Can1bridge
New York
Melbourne
Port Chester
Sydney
Published by the Press Syndicate of the University ofCambridge
The Pitt Building, Trumpington Street, Cambridge CB2 lRP
40 West 20th Street, New York, NYlOOll, USA
10, Stamford Road, Oakleigh, Melbourne 3166, Australia
©
Cambridge University Press 1991
Printed
in
Great Britain at the University Press, Cambridge
British Library cataloguing in publication data available
Library
0/
Congress cataloging in publication data available
Contents
.
Xl
Preface -
please read!
...
XUI
A Question of Ternlinology
A Guide to Notation
XIV
1
Chapter 0: A Branching-Process Exalnple
0.0.
Introductory remarks.
0.1.
Typical number of children,
X.
0.2.
Size
of
nth
generation,
Zn.
0.3. Use of conditional expectations. 0.4. Extinction
probability,
1r.
0.5. Pause for thought: measure. 0.6. Our first martingale.
0.7. Convergence (or not) of expectations. 0.8. Finding the distribution of
MOC).
0.9.
Concrete example.
PART A: FOUNDATIONS
Chapter 1: Measure Spaces
14
1.0. Introductory remarks. 1.1. Definitions of algebra, a-algebra. 1.2. Ex-
amples. Borel a-algebras,
B(S),
8
=
B(R).
1.3. Definitions concerning
set functions. 1.4. Definition of measure space. 1.5. Definitions con-
cerning measures.
1.6.
Lemma. Uniqueness of extension, 7r-systems. 1.7.
Theorem. Caratheodory's extension theorem. 1.8. Lebesgue measure Leb
on
((0,1],8(0,1]). 1.9.
Lemma. Elementary inequalities.
1.10.
Lemma.
Monotone-convergence properties of measures.
1.11.
Example/Warning.
Chapter 2: Events
23
Model for experiment:
(n,
F,
P).
2.2.
2.1.
The intuitive meaning.
2.3.
Examples of
(n,F)
pairs.
2.4.
Almost surely (a.s.)
2.5.
Reminder:
lim sup
En, (En,i.o.).
2.7.
limsup,liminf,l lim, etc.
2.6.
Definitions.
v
Contents
vi
First Borel-Cantelli Lemma (BC1). 2.8. Definitions. lim inf
En, (En,
ev).
2.9. Exercise.
29
Chapter 3: Random Variables
3.1. Definitions. E-measurable function, mE, (mE)+ ,bE. 3.2. Elementary
Propositions on measurability. 3.3. Lemma. Sums and products of mea-
surable functions are measurable. 3.4. Composition Lemma. 3.5. Lemma
on measurability of infs, liminfs of functions. 3.6. Definition. Random
variable. 3.7. Example. Coin tossing. 3.8. Definition. u-algebra generated
by a collection of functions on
n.
3.9. Definitions. Law, Distribution Func-
tion. 3.10. Properties of distribution functions. 3.11. Existence of random
variable with given distribution function. 3.12. Skorokod representation of
a random variable with prescribed distribution function. 3.13. Generated
u-algebras - a discussion. 3.14. The Monotone-Class Theorem.
Chapter
4:
Independence
38
4.1. Definitions of independence. 4.2. The 1r-system Lemma; and the
more familiar definitions. 4.3. Second Borel-Cantelli Lemma (BC2). 4.4.
Example. 4.5. A fundamental question for modelling. 4.6. A coin-tossing
model with applications. 4.7. Notation: UD RVs. 4.8. Stochastic processes;
Markov chains. 4.9. Monkey typing Shakespeare. 4.10. Definition. Tail
u-
algebras. 4.11. Theorem. Kolmogorov's 0-1 law. 4.12. Exercise/Warning.
Chapter
5:
Integration 49
5.0. Notation, etc.
p(/)
:=:
J
Idp, p(/;
A).
5.1. Integrals of non-negative
simple functions,
SF+.
5.2. Definition of
1l(/),
1
E
(mE)+. 5.3. Monotone-
Convergence Theorem (MaN). 5.4. The Fatou Lemmas for functions (FA-
TaU). 5.5. 'Linearity'. 5.6. Positive and negative parts of
I.
5.7. Inte-
grable function,
£1(8,
E,Il).
5.8. Linearity. 5.9. Dominated Convergence
Theorem (DaM). 5.10. Scheffe's Lemma (SCHEFFE). 5.11. Remark on
uniform integrability. 5.12. The standard machine. 5.13. Integrals over
subsets. 5.14. The measure
Ill,
1
E
(mE)+.
Chapter
6:
Expectation
58
Introductory remarks. 6.1. Definition of expectation. 6.2. Convergence
theorems. 6.3. The notation E(X;
F).
6.4. Markov's inequality. 6.5.
Sums of non-negative RVs. 6.6. Jensen's inequality for convex functions.
6.7. Monotonicityof
£P
norms. 6.8. The Schwarz inequality. 6.9.
£2:
Pythagoras, covariance, etc. 6.10. Completeness of
£P
(1
<
P
<
00).
6.11.
Orthogonal projection. 6.12.
The 'elementary formula' for expectation.
6.13. Holder from Jensen.
Contents
Vll
71
Chapter 1: An Easy Strong Law
7.1.
'Independence means multiply' - again!
7.2.
Strong Law - first version.
7.3.
Chebyshev's inequality.
7.4.
Weierstrass approximation theorem.
75
Chapter 8: Product Measure
8.0.
Introduction and advice.
8.1.
Product measurable structure,
El
X
E
2
•
8.2. Product measure, Fubini's Theorem. 8.3. Joint laws, joint pdfs.
8.4.
Independence and product measure.
8.5.
8(R)n = 8(Rn).
8.6.
The n-fold
extension.
8.7.
Infinite products of probability triples.
8.8.
Technical note
on the existence of joint laws.
PART B: MARTINGALE THEORY
83
Chapter
9:
Conditional Expectation
9.1.
A motivating example.
9.2.
Fundamental Theorem and Definition
(Kolmogorov,
1933). 9.3.
The intuitive meaning.
9.4.
Conditional ex-
pectation as least-squares-best predictor.
9.5.
Proof of Theorem
9.2. 9.6.
Agreement with traditional expression.
9.7.
Properties of conditional ex-
pectation: a list.
9.8.
Proofs of the properties in Section
9.7. 9.9.
Regular
conditional probabilities and pdfs.
9.10.
Conditioning under independe.nce
assumptions.
9.11.
Use of symmetry: an example.
Chapter
10:
Marthlgales
93
10.1.
Filtered spaces.
10.2.
Adapted processes.
10.3.
Martingale, super-
martingale, submartingale.
10.4.
Some examples of martingales.
10.5.
Fair
and unfair games.
10.6.
Previsible process, gambling strategy.
10.7.
A fun-
damental principle: you can't beat the system!
10.8.
Stopping time.
10.9.
Stopped supermartingales are supermartingales.
10.10.
Doob's Optional-
Stopping Theorem.
10.11.
Awaiting the almost inevitable.
10.12.
Hitting
times for simple random walk.
10.13.
Non-negative superharmonic func-
tions for Markov chains.
Chapter
11:
The Convergence Theorenl
106
11.1.
The picture that says
it
all.
11.2.
Up crossings.
11.3.
Doob's Upcross-
ing Lemma.
11.4.
Corollary.
11.5.
Doob's 'Forward' Convergence Theorem.
11.6.
Warning.
11.7.
Corollary.
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