Book:David Williams/Probability with Martingales

Subject Matter

 * Probability Theory

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