Probability: The Classical Limit Theorems

Henry McKean’s new book Probability: The Classical Limit Theorems packs a great deal of material into a moderate-sized book, starting with a synopsis of measure theory and ending with a taste of current research into random matrices and number theory. The book ranges more widely than the title might suggest.

The classical limit theorems of probability — the weak and strong laws of large numbers (LLN), the central limit theorem (CLT), the law of the iterated logarithm, and the arcsine law — are featured prominently, occurring first in their simplest forms and then in progressively more sophisticated settings. The reader starts with Bernoulli random variables and then is taken on a tour through random walks, Markov chains, Brownian motion, and ergodic theory. Along the way are excursions into physics, differential equations, geometry, and continued fractions.

If one were to split McKean’s book into two volumes, a natural place to do so would be after the chapter on ergodic theory. This chapter ends with a discussion of the LLN and CLT for geodesic flows. After this the book turns to topics not as directly related to the classical limit theorems. The final chapters are devoted to information and communication theory, random matrices, and especially statistical mechanics.

Probability: The Classical Limit Theorems is pleasant to read, containing generous expository discussion and historical notes. Give the ambitious scope of the book, the development is necessarily quite brisk. Someone already familiar with probability theory would enjoy reading the book for McKean’s perspective and applications. A student would find it hard to learn probability theory from this book alone. On the other hand, the same student would find this book complementary to the usual texts that are longer on detail but shorter on exposition. McKean’s reference early on to a measure theory book by M. E. Munroe sets the tone of the book:

Munroe [1953] tells you most of what you need — and nothing more, which I like.

There are numerous exercises sprinkled throughout the book. Most of these are exhortations to fill in details left out of the main discussion or illustrative examples. The exercises are a natural part of the book, unlike the exercises in so many books that were apparently grafted on after-the-fact at a publisher’s insistence.

McKean has worked in probability and related areas since obtaining his PhD under William Feller in 1955. His book contains invaluable insights from a long career.

John D. Cook is the founder of Singular Value Consulting and blogs regularly at The Endeavour.