David Williams Probability with Martingales
This book, which is essentially the set of lecture notes for a third-year undergraduate course at Cambridge University, is a nice textbook on measure-theoretic probability theory. It is a rigorous and self-contained textbook. After some intuitive introduction, it defines most probability concepts (event, probability, random variable, expectation, etc.) in a rigorous measure-theoretic language, it states many useful key results from measure theory in the main text, and gives complete proofs of these results in appendices. This book is a modern textbook. It has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's strong law of large numbers and the three-series theorem by martingale techniques, and the central limit theorem via the use of characteristic functions. This book also contains many lively examples and interesting applications. For the applications of martingale theory, it discusses the discrete Black-Scholes formula, the Kalman-Bucy filter, etc. Another feature is the arrangement of exercises. According to the author, ``The most important chapter in this book is Chapter E: Exercises'', which consists of the homework sheets given by the author to his students. The author attempts to train the creativeness of his readers; he likes readers to first read the statement of a result, and then to try to prove it for themselves before they read the proof given in the appendices. This book is also compact; including references it contains only 251 pages. Besides Chapter 0 (A branching-process example (pp. 1–13)), the book is divided into four parts. A description of the chapter headings is as follows: Part A: Foundations. Chapter 1: Measure spaces (pp. 14–22). Chapter 2: Events (pp. 23–28). Chapter 3: Random variables (pp. 29–37). Chapter 4: Independence (pp. 38–48). Chapter 5: Integration (pp. 49–57). Chapter 6: Expectation (pp. 58–70). Chapter 7: An easy strong law (pp. 71–74). Chapter 8: Product measure (pp. 75–82). Part B: Martingale theory. Chapter 9: Conditional expectation (pp. 83–92). Chapter 10: Martingales (pp. 93–105). Chapter 11: The convergence theorem (pp. 106–109). Chapter 12: Martingales bounded in L2 (pp. 110–125). Chapter 13: Uniform integrability (pp. 126–132). Chapter 14: UI martingales (pp. 133–142). Chapter 15: Applications (pp. 153–171). Part C: Characteristic functions. Chapter 16: Basic properties of CFs (pp. 172–178). Chapter 17: Weak convergence (pp. 179–184). Chapter 18: The central limit theorem (pp. 185–191). The last part is Appendices; it contains Chapter E: Exercises (pp. 224–244) and the appendices for most chapters. In these appendices, rigorous proofs of the results stated in the main text are given.
Reviewer: Wang, Jia Gang [form MathSciNet]