# zbMATH — the first resource for mathematics

Introduction to probability theory. Transl. from the Japanese. (English. Japanese original) Zbl 0545.60001
Cambridge etc.: Cambridge University Press. X, 213 p. hbk: £15.00; pbk: £6.00 (1984).
The author is one of the best known probability theorists who has given lectures at Kyoto University, Aarhus University and at Cornell University. Based on these courses he wrote a book ”Probability theory” (in Japanese), whose first four chapters have been translated to English and published as ”Introduction to probability theory”.
Chapter 1 of this book deals with finite situations, but from an advanced standpoint. Introducing mixing, direct and tree compositions the author presents in a uniform way conditional probabilities and independence. Chapter 2 and 3 discuss probability measures and fundamental concepts in the advanced probabiliy theory. At the beginning of the second chapter he reminds some fundamental facts from measure theory through a number of exercises. Then, the extension theorem of probability measures, direct product of probability measures, convergence of distributions and characteristic functions are explained. In chapter 3 the author defines events, random variables, independence, conditioning and so on. The conditional probability is first introduced with respect to decomposition of the sample space (Kolmogorov’s definition) and then with respect to $$\sigma$$-algebra of subsets of the sample space (Doob’s definition). Finally, chapter 4 deals with properties of infinite sums of independent real random variables. Among others the author studies the strong law of large numbers, central limit theorems, the law of iterated logarithm and Poisson’s law of rare events.
Summarizing, this book is particularly valuable to students taking courses in probability theory who need a modern introduction to the subject.
Reviewer: W.Szpankowski

##### MSC:
 60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 60A10 Probabilistic measure theory 60Fxx Limit theorems in probability theory