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Probabilities on Algebraic Structures

Ulf Grenander
Dover Publications
Publication Date: 
Number of Pages: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Ittay Weiss
, on

Probability theory and algebraic structures form, each, a formidable area of research, with plenty of theoretical investigations carried out through the years as well as numerous applications to many areas of science. As one would expect, books devoted to teaching and investigating probability theory and abstract algebra at various levels are plentiful. But that is so only as long as one is interested in either probability theory or algebraic structures, not both at the same time. Somewhat surprisingly, the study of probabilities on algebraic structures, which, of course, suitably interact with the algebra, is much more limited. One may wrongly conclude that such an endeavor is perhaps not fruitful and the book does a wonderful job in setting things straight.

The book is at an advanced level, requiring a substantial understanding of analysis but only a very shallow acquaintance with algebraic structures. The topics covered are stochastic semi-groups, stochastic groups (compact as well as locally compact), stochastic Lie groups, stochastic linear spaces, and stochastic algebras. Certainly not a book for a light reading on the beach, but rather an intense investigation of a beautiful and young area of some neglect — a powerful fusion between algebra and probability theory. The theme is an investigation of what happens to the classical results (primarily limit theorems, law of large numbers, etc.) on random variables on the reals, when the reals are replaced by other algebraic structures.

The author did not include any preliminary chapters, and the result is thus a rather short book concentrating on the core subject matter, a sort of a survey of results in a relatively new area where no decisive expository text has yet emerged. The student interested in reading the book will thus have to accumulate the necessary background material (a healthy dose of functional analysis and harmonic analysis) elsewhere. The exposition has a certain dramatic feel to it which I found quite captivating and thoroughly enjoyable in this extremely well-written book which offers a valuable addition at the advanced level.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.


1. Historical background and practical motivation of the problem
2. Stochastic semi-groups
3. Stochastic groups; compact and commutative cases
4. Stochastic Lie groups
5. Locally compact stochastic groups
6. Stochastic linear spaces
7. Stochastic algebras