Exercises in Probability: A Guided Tour from Measure Theory to Random Processes,via Conditioning

Loïc Chaumont and Marc Yor

Cambridge University Press, 2012, xx + 279 pages, $51.00, paperback

ISBN: 978-1-107-60655-5

Readership: Statistics graduate students, mathematics graduate students, researchers in probability and statistics.

The book is the second edition of an interesting collection of 120 problems presented by accomplished and renowned researchers on probability theory. The book is presented at an advanced measure theoretic level. The choice of the problems reflect the authors’ research interests and is evident from their acknowledgement in the introduction: “…we do not view this set of exercises as being ‘the’ good companion to a course in probability theory …, but rather we have tried to present some perhaps not so classical aspects...”. The collection, nonetheless, provides a profound insight into some of the more difficult techniques that have their place in the arsenal of a modern probabilist. In conjunction with the detailed solutions and relevant references, the book is a valuable educational resource for a beginning researcher in probability and mathematical statistics.

There are six chapters each containing a set of problems. The themes logically evolve from fundamentals and distributional aspects to theory of stochastic processes. From a general perspective the problems are, as admitted by the authors, of two different sorts: the ‘stripping’ exercises and the ‘dressing’ exercises. The former attempt to bring advanced reasoning needed to pursue a theoretical investigation to the level accessible to a student with a modest exposure in fundamentals of measure theory based probability. The latter set of exercises aim towards the opposite direction where some basic facts that are typically one-dimensional in their initial scope are embedded in an infinite-dimensional framework. Essentially all problems are accompanied by their complete solutions. Two exercises (2.7 and 6.29) that contain some unsolved questions (marked in the text by circles) are presented as challenge problems to the readers.

Several simple but extremely useful additions facilitate efficient usage of the text. These include a short list of frequently used notation and comments following individual problems that provide additional insight. Pointers to additional citation is also a useful resource for readers interested in further contextual learning.