A Second Course in Probability      by Ross, Sheldon M.; Peköz, Erol A.

Ross (Univ. of Southern California) and Pekoz (Boston Univ.) present a delightful, very accessible book with advanced topics in probability without going through classical and advanced probability theory literature. A salient feature is that advanced probability theory topics are covered without needing much knowledge in real analysis. The book is intended for second courses in probability for undergraduate as well as graduate students in mathematics, engineering, finance, statistics, and actuarial sciences. Seven chapters begin with brief chapter overviews and discuss topics such as measures theory, limit theorems, coupling and Stein's method, expectations and bounding probabilities, martingales, Markov Chains, renewal theory, and Brownian motion. The proof of the law of large numbers is presented using the ergodic theorem, and the central limit theorem is proved using Stein's method and Brownian motion embeddings. The only background required is familiarity with calculus and an undergraduate course in probability, since all other concepts (e.g., Lebesgue integral, convergence, etc.) are introduced and explained fully as needed. A noteworthy feature is the inclusion of a wealth of solved examples. Each chapter ends with a set of exercises for readers to explore. Useful index; short reference list. Summing Up: Highly recommended. Upper-division undergraduates; graduate students. D. V. Chopra Wichita State University