Probability-1

This book is axiomatic and abstract, and presents a comprehensive but pure-math approach to probability theory. The book is not applied at all; the examples are synthetic, and there’s almost no mention of statistics. The author belongs to the Kolmogorov school and follows their axiomatic approach. This is an English translation of the fourth Russian edition and is the first of two volumes. The second volume was projected to be published in 2016 but has not appeared or been announced yet.

The present book comprises the first three chapters of the Russian edition. Chapter 1 is titled “Elementary Probability Theory” but in this context “elementary” means that it deals only with finite event spaces; it covers a lot of advanced topics. It uses mostly finite methods, although there is also some continuous math regarding limits, such as the law of large numbers and some other early limit theorems regarding Bernoulli distributions. The main example areas, besides Bernoulli, are martingales and Markov chains. The second chapter fills in a lot of the background needed for continuous distributions. A lot of it looks just like a course in measure and integration, except that everything is expressed in probability rather than measure terms, and there’s an emphasis on the Lebesgue-Stieltjes rather than the Lebesgue integral. The third chapter deals mostly with limit theorems, especially the Central Limit Theorem, and various kinds of convergence in measure. The second volume will cover sums of random variables and a number of specialized topics.

There are good exercises throughout the book, although skimpy. There is a companion problems book keyed to this one, Shiryaev’s Problems in Probability.

The translation reads smoothly, and is in good idiomatic English. The only glitch I noticed was that the French mathematician Vandermonde appears as Wandermonde (p. 144), presumably a victim of two levels of transliteration.

I think this is a good book if you like this kind of very abstract approach. Feller’s very venerable An Introduction to Probability Theory and Its Applications is a good alternative if you like something more concrete. Feller’s volume 1 covers the discrete cases very thoroughly, and volume 2 does a good job on continuous distributions and limit theorems, although it avoids most of the measure machinery and takes a more classical-analysis approach.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.