A Graduate Course in Probability

Perhaps not the Tucker best-known to mathematicians, Howard G. Tucker was professor at the University of California, Irvine, where this text was used in a two-semester, second-year graduate course. Prerequisites include a measure-theoretic real analysis course comparable to, say, Royden and Fitzpatrick. The author lists as assumed the Lebesgue integral and Lebesgue dominated convergence theorem, as well as the theorems of Fubini, Radon-Nikodym, and Egorov. No prior course in probability is required, though it would certainly help the student to see what the theory presented here might be used for. An informal survey of Ph.D. statisticians indicated that such a course was a common, though far from universal, part of a Ph.D. program in statistics. However, this is clearly a mathematics textbook that discusses mathematics useful in advanced probability, and not a statistics textbook.

The book covers exactly the material covered in the underlying course with few digressions or applications. It opens with three chapters on probability spaces, distributions, and independence. There follows three chapters on limit theorems such as the central limit theorem. A seventh chapter discusses conditional expectation and martingales. The final chapter covers stochastic processes such as Brownian motion. The presentation is strictly theorem-proof with minimal text that helps the reader follow the meaning of theorems and the logical flow of their proofs. Those proofs are reasonably detailed. A moderate number of student exercises nearly always ask for a proof. It is the author’s intent that students do all of the exercises.

This appears to be a viable candidate for use in any course that matches its content and level. Students are sure to appreciate its bargain price.

After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at statistics.com and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.