# Book:David Williams/Probability with Martingales

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## David Williams:

## 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**