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A Probability Path

Sidney I. Resnick
Publication Date: 
Number of Pages: 
Modern Birkhäuser Classics
[Reviewed by
William J. Satzer
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This introduction to measure-theoretic probability is intended for students whose primary interest is not mathematics but statistics, engineering, biology, or finance. The book is a welcome reprint in paperback of the 2005 edition of a book that was first published in 1998. The author felt there was a demand for a book motivated by applications that could be used to convey the essentials in a one semester course. He says that mathematicians writing books like this tend to write for students and other mathematicians who come to the subject for its beauty, not for its applicability. (Perhaps a rather overbroad generalization.)

The book’s pace, according to the author, is “quick and disciplined”. He’s not kidding. This is by no means a watered-down introduction for students who couldn’t handle a more rigorous course. It would be quite suitable for graduate students in mathematics too, except that the measure theory is so tightly integrated with the theory of probability that students would not see it in the more abstract context they would typically get in a graduate real analysis course.

The book begins with three chapters that develop measure theory in sufficient depth that students could then independently pursue and understand more advanced material. After that, the author introduces independence and expectation (along with the Lebesgue integral), and then — at some length — carefully sorts out all the different categories of convergence. The payoff comes in the final four chapters: the law of large numbers, convergence in distribution, the central limit theorem, and then a long piece on martingales.

This all comes with notably good sets of exercises (which the author suggests that students be “encouraged or even forced to do”) and an occasional dose of the author’s sly humor: “A martingale is a stochastic process … used to model a fair sequence of gambles (or, as we say today, investments).”

Those who know the author’s Adventures in Stochastic Processes will be disappointed to learn that Happy Harry who appeared in several exercises in that book is absent here. The author says he’s on vacation.

Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1 Sets and Events

2 Probability Spaces

3 Random Variables, Elements and Measurable Maps

4 Independence

5 Integration and Expectation

6 Convergence Concepts

7 Laws of Large Numbers and Sums of Independent Random Variables

8 Convergence in Distribution

9 Characteristic Functions and the Central Limit Theorem

10 Martingales