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Probability. (English) Zbl 0784.60001

Springer Texts in Statistics. New York, NY: Springer-Verlag. xvii, 282 p. (1993).
Fifty years ago “calculs de probabilitiés” still belong to the department of physics and are mostly used as a tool to control errors in experimental measures. Today it is considered as an important branch of applied mathematics whose “stochastic concepts” on the one hand unify the mathematical foundations of statistics and probabilities, and on the other hand open many important problems connected with other branches of pure mathematics such as “stochastic differential equations on manifolds”, “calculus of variations in Wiener spaces”, “potential theory of Markov processes”,... In this situation the way of writing a textbook at the introductory graduate level depends both on the orientation one would like to aim the students at (such as either doing applications in engineering or biostatistics or doing researches in mathematical sciences) and on the taste of the writer. One may think about the classical “Measure theory” of P. R. Halmos (1950; Zbl 0040.168), where probability theory is treated easily as a “measure space with total measure one”, or the classical “Mathematical foundations of the calculus of probability” of J. Neveu (1965; Zbl 0137.113), where probability theory turns out to be a special chapter of real and functional analysis.
The author of the textbook under review takes a moderate point of view as explained in his Preface: “This is a book to teach from. It is not encyclopaedic, and may not be suitable for all references purposes”. Therefore, “students who master this text should be able to read the ‘hard’ books on probability with relative ease, and to proceed to further study in statistics and stochastic processes”. In other words, this is a book on probability in the same size but in different level and taste in comparing with, for instance, “A course in probability theory” of K. L. Chung (1974; Zbl 0345.60003), where the author treats “random variables” (instead of events) as the main basic objects in his approach of stochastic concepts.
The main body of the book consists of the first six chapter, which are respectively Chap. 1 (Probability), Chap. 2 (Random variables), Chap. 3 (Independence), Chap. 4 (Expectation), Chap. 5 (Convergence of sequences of r.v.) and Chap. 6 (Characteristic functions). Besides of Chap. 7 on classical limit convergence theorems which is important in itself, the reader could read independently with the Chapters 6 and 7 the next Chapters 8 (on conditional expectation as projections in \(L^ 2)\) and 9 (on martingales).
Advanced technics on “probabilistic computations” which may discourage the reader in his first reading are warned by the author as “technical asides”, “complements” or referred to the “exercises”.
The book is in general well written, probabilistic concepts are usually motivated intuitively by examples, then consisely defined by mathematical terminologies, theorems are clearly stated; however although one does not always expect a mathematical rigor in their proofs, the different chapters are well organized in a unified way. It is recommendable as a good introduction book on probability theory for students in statistics, biostatistics, engineering, economics or applied mathematics in general with the reserve that students must not expect to find out here the good sources of immediate applications for their specialities.
Reviewer: X.L.Nguyen (Hanoi)

MSC:

60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60Fxx Limit theorems in probability theory
60Gxx Stochastic processes
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