Probability with Martingales

Citation

Williams, D. (1991). Probability with Martingales. Cambridge University Press (Cambridge Mathematical Textbooks).

Chapter Summary

The book is, in the author's words, essentially the lecture notes for a third-year undergraduate course at Cambridge. It teaches the rigorous, measure-theoretic theory of probability and uses martingales as the thread that carries the reader from the foundations to the strong law of large numbers and beyond. The text has a motivating Chapter 0, three Parts (A: Foundations, B: Martingale Theory, C: Characteristic Functions), a set of appendices that hold the longer measure-theoretic proofs, and Chapter E, a chapter of exercises. Within chapters, sections are numbered and several results are given names and marked with arrows to show their importance.

Chapter 0: A Branching-Process Example

A motivating example before any theory. The chapter follows the number of children, the size of the nth generation and the probability of extinction of a branching process, pauses to ask what measure is needed, and introduces the first martingale and the question of whether expectations converge.

Chapters 1–3: Measure Spaces; Events; Random Variables

Part A begins with σ-algebras, Borel σ-algebras, measure spaces, the π-system uniqueness lemma, Carathéodory's extension theorem and Lebesgue measure. It then gives the (Ω, F, P) model for an experiment, almost sure statements, lim sup and lim inf of events and the first Borel–Cantelli lemma, and defines random variables as measurable functions, with laws, distribution functions, the Skorokhod representation and the Monotone-Class Theorem.

Chapters 4–6: Independence; Integration; Expectation

Independence is defined through π-systems and tied to the second Borel–Cantelli lemma, a coin-tossing model, Markov chains and Kolmogorov's 0–1 law. Integration with respect to a measure builds the Monotone-Convergence Theorem, Fatou's Lemmas, Dominated Convergence and Scheffé's Lemma. Expectation then gives the standard inequalities (Markov, Jensen, Hölder, Schwarz), the L^p spaces and orthogonal projection in L².

Chapters 7–8: An Easy Strong Law; Product Measure

An early strong law of large numbers, with Chebyshev's inequality and the Weierstrass approximation theorem, shows the foundations at work. Product measure, Fubini's Theorem, joint laws and densities, and infinite products of probability triples complete Part A.

Chapters 9–11: Conditional Expectation; Martingales; The Convergence Theorem

Part B begins with conditional expectation, defined by Kolmogorov's fundamental theorem and interpreted as a least-squares-best predictor. Martingales, supermartingales, adapted and previsible processes, stopping times and Doob's Optional-Stopping Theorem follow, with hitting times for random walk as an application. Chapter 11 proves Doob's Upcrossing Lemma and the forward convergence theorem, which the notation guide marks as the Martingale Convergence Theorem.

Chapters 12–14: Martingales Bounded in L²; Uniform Integrability; UI Martingales

Square-integrable martingales yield Kolmogorov's three-series theorem, the strong law under variance conditions and the angle-brackets process. Uniform integrability gives a necessary and sufficient condition for L¹ convergence. UI martingales then deliver Lévy's Upward and Downward Theorems, martingale proofs of Kolmogorov's 0–1 law and the strong law, Doob's inequalities, Kakutani's theorem on product martingales and the Radon–Nikodým theorem.

Chapter 15: Applications

A short chapter of applications: a martingale-representation result, option pricing and the discrete Black–Scholes formula, the Mabinogion sheep problem, the Kalman–Bucy filter and "harnesses".

Chapters 16–18: Basic Properties of CFs; Weak Convergence; The Central Limit Theorem

Part C, which does not use martingales, covers characteristic functions and Lévy's inversion formula, weak convergence with Skorokhod representation and tightness, and the Central Limit Theorem through Lévy's convergence theorem.

Appendices and Chapter E: Exercises

The appendices A1, A3, A4, A5, A9, A13, A14 and A16 prove results from the matching chapters, including Carathéodory's theorem, the Monotone-Class Theorem and the Law of the Iterated Logarithm. Chapter E occupies about nineteen pages.

Key Concepts

Chapters 1–8: Foundations

• σ-algebra and measure: the collection of events and the countably additive function that assigns them size.
• Borel–Cantelli lemmas: criteria for an event to occur infinitely often, in the cases without and with independence.
• Monotone Convergence, Fatou and Dominated Convergence: the three results that justify exchanging limits and integrals.

Chapters 9–11: Conditional Expectation and Martingales

• Conditional expectation: the best prediction of a random variable given a σ-algebra of information.
• Martingale and stopping time: a fair-game process and a random time decided by information so far; Doob's Optional-Stopping Theorem links them.
• Doob's Upcrossing Lemma: the tool that proves martingale convergence.

Chapters 12–15: Convergence Theory

• Uniform integrability: the condition under which convergence in probability gives L¹ convergence.
• Lévy's Upward and Downward Theorems: convergence of conditional expectations along an increasing or decreasing filtration.

Chapters 16–18: Characteristic Functions

• Characteristic function: the Fourier transform of a law, which determines it and behaves well under sums of independent variables.
• Weak convergence and tightness: convergence in law and the compactness condition that ensures subsequential limits.

Critical Analysis

Strengths

• Martingales as the organising idea. The preface says the book exists to make the rigorous theory of probability lively, and the structure supports this: measure theory is developed as far as needed, and martingales then prove the strong law, Kolmogorov's 0–1 law and the Radon–Nikodým theorem. The zbMATH reviewer, U. Mansmann, describes Williams as presenting the martingale concept as a natural one that gives easy access to the main points of probability theory.
• Measure theory in proportion. The preface says that central results are stated axiomatically in the main text and proved carefully in appendices. Mansmann says the appendices are accessible to a newcomer.
• Exercises as the heart of the book. The preface calls Chapter E the most important chapter and says the exercises are the homework sheet Williams gives Cambridge students. It adds that the text contains many other easier exercises, which the reader creates by trying to prove a result before reading the proof.
• Candid about choices. "A Question of Terminology" explains why random variables are treated as functions rather than equivalence classes, and "A Guide to Notation" ranks results with arrows. The preface lists books for further study and says topics judged slightly too advanced were omitted.

Limitations

• Lecture-note format. The text is a set of lecture notes, and the preface itself lists other books for a more thorough treatment of many of the topics.
• Discrete time only. The preface expresses the hope that readers will go on to the theory where the parameter set is uncountable, such as continuous time, which the book does not cover.
• Applications are brief. Chapter 15 is short and its applications (option pricing, the Kalman–Bucy filter, harnesses) are first glimpses rather than a developed treatment.
• Part C stands apart. Characteristic functions and the Central Limit Theorem do not use martingales, so the book's unifying theme does not extend to its last third.

Where It Fits in the Course

This is the measure-theoretic book in the probability and statistics course. Blitzstein and Hwang's Introduction to Probability is the elementary prerequisite, and Mitzenmacher and Upfal's Probability and Computing treats probability through discrete, algorithmic examples. Casella and Berger's Statistical Inference and the mathematical statistics texts use probability for estimation and testing without measure theory, whereas Williams supplies the rigorous footing that those texts leave out. Probability and Statistical Inference (Mavrakakis and Penzer) is a graduate-level book that combines probability with inference; Williams is the more focused choice for the theory of conditional expectation and martingales.

For stochastic processes, the book is a natural preparation for Øksendal's Stochastic Differential Equations, which assumes conditional expectation and martingale convergence (and gives appendices on both). Shreve's Stochastic Calculus for Finance I treats martingales in discrete time without measure theory and is a gentler companion.

How to Use This Book

The book is for students who have met probability informally and have a first course in real analysis; Section 2.5 recalls lim sup and lim inf. The preface says it grew out of a Cambridge third-year course.

Read the Preface, "A Question of Terminology" and "A Guide to Notation", then Chapter 0. Take Part A at a steady pace, deferring the long appendix proofs (A1, A3) on a first reading. Chapters 5 and 6 are essential for what follows. The core of Part B is Chapters 9 to 11, culminating in the Martingale Convergence Theorem; Chapters 12 to 14 deepen it, and Chapter 15 is optional. Part C is independent of the martingale chapters and can be read after Chapter 8.

Do the exercises of Chapter E as you go, since the preface describes them as the book's most important chapter, and try to prove each result before reading the proof. The books named in the preface are suitable follow-ups for a broader view of probability and for continuous-time theory.