Book:David Williams/Probability with Martingales

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David Williams: Probability with Martingales

Published $\text {1991}$, Cambridge University Press


Subject Matter


Contents

Preface – please read!
A Question of Terminology
A Guide to Notation
PART A: FOUNDATIONS
Chapter 0: A Branching-Process Example
0.0 Introductory remarks.
0.1 Typical number of children, $X$.
0.2 Size of $n^{\text{th}}$ generation, $Z_n$.
0.3 Use of Conditional Expectations.
0.4 Extinction probability, $\pi$.
0.5 Pause for thought: measure.
0.6 Our first martingale.
0.7 Convergence (or not) of expectations.
0.8 Finding the distribution of $M_\infty$.
0.9 Concrete example.
Chapter 1: Measure Spaces
1.0 Introductory remarks.
1.1 Definitions of algebra, $\sigma$-algebra.
1.2 Examples. Borel $\sigma$-algebras, $\map \BB S$, $\BB = \map \BB \R$.
1.3 Definitions concerning set functions.
1.4 Definitions of measure space.
1.5 Definitions concerning measures.
1.6 Lemmas. Uniqueness of extension, $\pi$-systems.
1.7 Theorem. Carathéodory's extension theorem.
1.8 Lebesgue measure $\mathrm {Leb}$ on $\struct {\hointl 0 1, \map \BB {\hointl 0 1} }$.
1.9 Lemmas. Elementary inequalities.
1.10 Lemma. Monotone-convergence properties of measures.
1.11 Example/Warning.
Chapter 2: Events
2.1 Model for experiment: $\struct {\Omega, \FF, \mathbf P}$.
2.2 The intuitive meaning.
2.3 Examples of $\struct {\Omega, \FF}$ pairs.
2.4 Almost surely (a.s.)
2.5 Reminder: $\limsup$, $\liminf$, $\downarrow \lim$, etc.
2.6 Definitions. $\limsup E_n$, ($E_n \text{ i.o.}$).
2.7 First Borel-Cantelli Lemma (BC1).
2.8 Definitions. $\liminf E_n$, ($E_N \text{ ev}$).
2.9 Exercise.
Chapter 3: Random Variables
3.1 Definitions. $\Sigma$-measurable function, $\mathrm m \Sigma$, $\paren {\mathrm m \Sigma}^+$, $\mathrm b \Sigma$.
3.2 Elementary Propositions on measurability>
3.3 Lemma. Sums and products of measurable functions are measurable.
3.4 Composition Lemma.
3.5 Lemma on measurability of $\inf$s, $\liminf$s of functions.
3.6 Definition. Random variable.
3.7 Example. Coin tossing.
3.8 Definition. $\sigma$-algebra generated by a collection of functions on $\Omega$.
3.9 Definitions. Law, Distribution Function.
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 $\sigma$-algebras – a discussion.
3.14 The Monotone-Class Theorem.
Chapter 4: Independence
4.1 Definitions of independence.
4.2 The $\pi$-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: IID RVs.
4.8 Stochastic processes; Markov chains.
4.9 Monkey typing Shakespeare.
4.10 Definition. Tail $\sigma$-algebras.
4.11 Kolmogrov's $0$-$1$ law.
4.12 Exercise/Warning.
Chapter 5: Integration
5.0 Notation, etc. $\map \mu :=: \int f d \mu$, $\map \mu {f ; A}$.
5.1 Integrals of non-negative simple functions, $SF^+$.
5.2 Definition of $\map \mu f$, $f \in \paren {\mathrm m \Sigma}^+$.
5.3 Monotone-Convergence Theorem (MON).
5.4 The Fatou Lemmas for functions (FATOU).
5.5 'Linearity'.
5.6 Positive and negative parts of $f$.
5.7 Integral function, $\map {\LL^1} {S, \Sigma, \mu}$.
5.8 Linearity.
5.9 Dominated Convergence Theorem (DOM).
5.10 Scheffé's Lemma (SCHEFFÉ).
5.11 Remark on uniform integrability.
5.12 The standard machine.
5.13 Integrals over subsets.
5.14 The measure $f \mu$, $f \in \paren {\mathrm m \Sigma}^+$.
Chapter 6: Expectation
Introductory remarks.
6.1 Definition of expectation.
6.2 Convergence theorems.
6.3 The notation $\expect {X ; F}$.
6.4 Markov's inequality.
6.5 Sums of non-negative RVs.
6.6 Jensen's inequality for convex functions.
6.7 Monotonicity of $\LL^p$ norms.
6.8 The Schwarz inequality.
6.9 $\LL^2$: Pythagoras, covariance, etc.
6.10 Completeness of $\LL^p$ ($1 \le p < \infty$).
6.11 Orthogonal projection.
6.12 The 'elementary formula' for expectation.
6.13 Hölder from Jensen.
Chapter 7: 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.
Chapter 8: Product Measure
8.0 Introduction and advice.
8.1 Product measurable structure, $\Sigma_1 \times \Sigma_2$.
8.2 Product measure, Fubini's theorem.
8.3 Joint laws, joint pdfs.
8.4 Independence and product measure.
8.5 $\map \BB \R^n = \map \BB {\R^n}$
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
Chapter 9: Conditional Expectation
Chapter 10: Martingales
Chapter 11: The Convergence Theorem
Chapter 12: Martingales bounded in $\LL^2$
Chapter 13: Uniform Integrability
Chapter 14: UI Martingales
Chapter 15: Applications
PART C: CHARACTERISTIC FUNCTIONS
Chapter 16: Basic Properties of CFs
Chapter 17: Weak Convergence
Chapter 18: Central Limit Theorem
APPENDICES
Chapter A1: Appendix to Chapter 1
A1.1 A non-measurable subset $A$ of $S^1$.
A1.2 $d$-systems.
A1.3 Dynkin's lemma.
A1.4 Proof of Uniqueness Lemma 1.6.
A1.5 $\lambda$-sets: 'algebra' case.
A1.6 Outer measures.
A1.7 Carathéodory's Lemma.
A1.8 Proof of Carathéodory's Theorem.
A1.9 Proof of the existence of Lebesgue measure on $\struct {\hointl 0 1, \map \BB {\hointl 0 1} }$.
A1.10 Example of non-uniqueness of extension.
A1.11 Completion of a measure space.
A1.12 The Baire category theorem.
Chapter A3: Appendix to Chapter 3
A3.1 Proof of the Monotone-Class Theorem
A3.2 Discussion of generated $\sigma$-algebras.
Chapter A4: Appendix to Chapter 4
A4.1 Kolmogorov's Law of the Iterated Logarithm.
A4.2 Strassen's Law of the Iterated Logarithm.
A4.3 A model for a Markov chain.
Chapter A5: Appendix to Chapter 5
A5.1 Doubly monotone arrays.
A5.2 The key use of Lemma 1.10(a).
A5.3 'Uniqueness of integral'.
A5.4 Proof of the Monotone-Convergence Theorem.
Chapter A9: Appendix to Chapter 9
A9.1 Infinite products: setting things up.
A9.2 Proof of A9.1(e).
Chapter A13: Appendix to Chapter 13
A13.1 Modes of convergence: definitions.
A13.2 Modes of convergence: relationships.
Chapter A14: Appendix to Chapter 14
A14.1 The $\sigma$-algebra $\FF_T$, $T$ a stopping time.
A14.2 A special case of OST.
A14.3 Doob's Optional-Sampling Theorem for UI martingales.
A14.4 The result for UI submartingales.
Chapter A16: Appendix to Chapter 16
A16.1 Differentiation under the integral sign.
Chapter E: Exercises
References
Index
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