This is a graduate level book in probability (the first four chapters) and stochastic processes (the remaining five chapters). The author covers a considerable amount of material in the 550 pages. The choice of topics and the style of presentation make the book valuable both as a classroom textbook and as a reference book. The target reader is the graduate student who has had a Probability class before (and hopefully some theory of integration, but this is not absolutely necessary) and also the general mathematician interested in the subject.
The first three chapters might even be thought of as a short course on analysis. They provide a concise exposition of measure theory and integration, Lp spaces, and convergence theorems, presented, of course with a view toward applications in probability. At the beginning of chapter II, on probability spaces, the author says that “implicit in the language are whole sets of attitudes, prejudices, and desires with which we hope to infect the reader.”
Chapter IV deals with conditional probability while chapters V through IX take the reader through the theory of stochastic processes.
The presentation style is effective and to the point. There are numerous exercises dispersed throughout the text, and the number of theorems is kept to a minimum, with only major results presented as such. The pace of the book, as well as the comments and historical references keep the reader interested.
The author makes definitive statements, such as “We describe two basic constructions: Ionescu-Tulcea’s and Kolmogorov’s. Together, they show the existence of all the probability spaces that were ever needed.” Peppered throughout the book are “integrating” pieces such as the paragraphs “On the Definitions” in chapter VII, or the “Markovian Bestiary” in chapter IX, which summarize and put in context the various concepts discussed. The book ends with a section titled “Notes and Comments,” which once again relates the presented material to the existing literature.
This is a very good exposition of the theory of probability and stochastic processes, modern yet not dry, and this reviewer warmly recommends it to the graduate student, to the mathematician working in related fields, and even to the adventurous undergraduate student.
Florin Catrina is Assistant Professor of Mathematics at St. John's University in Queens, New York.