Asymptotic Theory of Statistics and Probability, by Anirban DasGupta, New York, Springer, 2008, xxvii+722 pp., £55.99 or US$89.95 (hardback), ISBN 978-0-387-75970-8
Unlike many books on asymptotic theory, this book is set up to provide a broad collection of results on the topic without going into too much mathematical detail, and without providing proofs of those results. This allows the author to discuss an impressively wide variety of topics in both statistics and probabilistic theory. The eloquent style of writing makes this book very accessible as well as a joy to read. In particular, the introductions of each of the different topics are very effective: they set up the conceptual framework for the results that follow, and provide much valuable contextual insight in only one or two pages. After the introduction, each of the 34 regular chapters contains a selection of important results from the relevant theory, with many examples of their use, as well references to sources of more detailed information (the sources of proofs for the theorems presented are also separately listed at the front of the book). The chapters all end with a large number of exercises, as well as a bibliography of relevant references. Naming all the topics that are covered could easily fill a number of pages, and the interested reader is invited to check out the 12-page table of contents that is available on the publisher's website. However, Chapter 35 deserves special mentioning: it doesn't deal with asymptotic theory at all, but rather contains a large number of inequalities (in probability, but also linear algebra and analysis) that the author has collected over the years, each duly referenced. With this kind of encyclopedic book, an index is important, and fortunately it is well done, extending over 30 pages.
This book is strongly recommended for those interested in the use of asymptotics in statistics and probability problems. Even for those interested in more details, it provides a quick and accessible overview of the available results, providing both a basic understanding of their context and references to sources where a more detailed treatment can be found.