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Probability: A Graduate Course

Allan Gut
Publication Date: 
Number of Pages: 
Springer Texts in Statistics
[Reviewed by
Pietro Poggi-Corradini
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Most mathematics graduate programs in the US run an “Advanced Probability” course for second-year students, covering what is familiarly known as probability “with measure theory.” Having been asked to teach this course for the first time last year, I scanned various webpages of graduate programs around the country, to see which texts are the most commonly used. My brief survey brought up Rick Durrett’s Probability: Theory and Examples as the most popular choice by a long shot. I therefore chose Durrett’s book for the course. In this review I will try to determine how Gut’s book would fare as a possible alternative.

Even though almost every student in the class had had a real analysis course (there were a couple of engineers in the class who had not), I spent almost a third of the first semester reviewing measure theory. Conveniently, in Durrett’s recent edition the first chapter contains a fairly in-depth account of measure theory from a probabilistic point of view. Gut’s book on the other hand, begins with a very cursory introduction to basic measure theory and moves on much more quickly to probability per se. In retrospect, I think I spent too much time reviewing basic concepts. In addition, as I soon found out, Durrett’s book contains much more material than can be covered in a year. So skipping is inevitable. Were I to teach this material again, I would not cover the introductory chapter as thouroughly as I did.

Now what about probability theory proper? I am of the opinion that a graduate course in probability should cover a small number of “inalienable” topics, and that among these are the law of large numbers, the central limit theorem, martingales, Markov chains, and Brownian motion. Durrett’s book does this and much more. The challenge for the instructor is to know what advanced sections to skip so as to be able to stay on target. I ended up doing measure theory, the law of large numbers and the central limit theorem the first semester and martingales, Markov chains and Brownian motion the second semester.

Gut’s book does not cover Markov chains and Brownian motion, but instead stops at martingales and spends more time with the basics. For instance, several results that are exercises in Durrett are discussed in detail by Gut. This is not to say that the exercises in Durrett are overly difficult. Indeed, I think they are one of the strengths of the book. Still it’s nice that Gut takes the time to go into details about the basic theory.

Gut organizes things differently but mostly focuses on basic probability. So an instructor using Gut would not have to decide which more advanced topics to skip and could safely follow the text linearly. However, Gut does not cover two essential topics for a graduate probability course. It is especially worrying that Brownian motion, the jewel of probability theory, is left out. In view of the centrality of Brownian motion in modern-day probability and analysis, and in view of the fact that Brownian motion is both a martingale and a Markov chain, it seems to me that one cannot skip this in a graduate course. I’d say more: the whole course should have as ultimate goal and crowning achievement to introduce and discuss Brownian motion exploiting all the background that has been built up. This, of course, is my personal opinion as an outsider (I was trained as a complex analyst). Therefore, Gut’s book would be appropriate for the first semester, or the first two quarters (if on the quarter system), but one would then have to switch to some other text.

All in all, I would recommend Gut’s book as an alternative source to any student who has difficulties with the standard material.

Pietro Poggi-Corradini is Professor of Mathematics at Kansas State University.

​​Preface to the First Edition
Preface to the Second Edition
Outline of Contents
Notation and Symbols
Introductory Measure Theory
Random Variables
Characteristic Functions
The Law of Large Numbers
The Central Limit Theorem
The Law of the Iterated Logarithm
Limited Theorems
Some Useful Mathematics