The Theory of Probability Santosh S. Venkatesh Cambridge University Press 2013 805 pages hardcover ISBN 9781107024472 textbook advanced undergraduates and above Dr. Manuel Vogel, TU Darmstadt and GSI Darmstadt,
[email protected] Probability is by its nature a conceptually demanding topic rather than necessarily being mathematically difficult. It is hence not easy to convey at any level of mathematical abstraction. 'The Theory of Probability' is an effort to lead us into the world of probability and its concepts from the obvious to the non-obvious, which is also more or less the historical development of the field (as is of course also true for most other fields of science). In the present book, however, this has a particular meaning as it puts a special focus on the history of probability theory, which started with phenomenological description of chances in games as early as in the seventeenth century and took a turn in the 1930s when Kolmogorov introduced the axiomatic approach to probability, hence making the topic accessible with new mathematical instruments. But still, as the author points out, the field is far from being static with major advances having been made as recently as in the last quarter of the twentieth century and the expectation of more to come. The individual topics as presented in the book were originally a loose sequence of essays which have now been structured to allow a selfguided study. The preface goes to the effort of explaining the interconnections between topics and suggests certain ways of selecting and sequencing the material under different aspects, there is even a figure and a table explaining the layout and the use of the book. The mere sequence of chapters is 'Probability spaces', 'Conditional probability', 'A first look at independence', 'Probability sieves', 'Numbers play a game of chance', 'The normal law', 'Probabilities on the real line', 'The Bernoulli schema', 'The essence of randomness', 'The coda of the normal', 'Distribution functions and measure', 'Random variables', 'Great expectations', 'Variations on a theme of integration', 'Laplace transforms', 'The law of large numbers', 'From inequalities to concentration', 'Poisson approximation', 'Convergence in law', 'Selection theorems', 'Normal approximation', and an appendix with 'Sequences, functions, spaces'. Each chapter presents problems, altogether about 500 of them. Selected solutions are available on an accompanying website at www.cambridge.org/venkatesh. Even more helpful for a self-guided study are, however, the many examples and practical applications, which allow a more intuitive access to the material. The typical way of presentation is a motivation of a problem within its historical background, a resulting theorem which is proven (or methods of proof are discussed), and applications or examples, sometimes connected with homework problems. The material is presented in a way which allows to follow the historical development of the field. The text is well-written and although the topics are discussed with all mathematical rigour, it usually does not exceed the capabilities of an advanced undergraduate student. The author, Santosh Venkatesh, is associate professor at the University of Pennsylvania and has won a distinguished award for teaching. One can indeed see in the way the text is written that the author enjoys presenting the material and teaching us so that we understand. Altogether, the book is very insightful, both with respect to probability theory being presented in a mathematically rigorous way, and with respect to historical detail. It can be recommended without constraint as a textbook for advanced undergraduates, but also as a reference and interesting read for experts.
