Featured Review:; Introduction to Probability and Its Applications

Abstract

SIAM Review, Volume 68, Issue 1, Page 207-208, February 2026.
The two volumes of Feller’s Introduction to Probability are classic and comprehensive. They span a huge range of topics, starting from measure theory and basic probability distributions and moving on to topics such as Markov chains and the central limit theorem. The books are very well written and accessible. It seems that most of the theory that is being employed in 2025 by applied modelers and analysts is touched on in these books in some way. Of course, the emphases are different. Much of the foundational material on probability theory has not changed a lot since Feller’s two volumes, including basic measure theory, the law of large numbers and the central limit theorem, conditional probabilities, and theorems such as the Borel Cantelli Lemma. In contrast, it seems that Feller spends a lot of time surveying a range of special probability models and distributions whereas more modern treatments would perhaps spend more time in surveying general analytic techniques. This is probably partly due to the fact that numerical simulation techniques were not nearly as powerful when the book was written, so scholars tended to focus more on specific tractable models.