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Probability Theory: A Comprehensive Course

Achim Klenke
Publication Date: 
Number of Pages: 
[Reviewed by
Mehdi Hassani
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Since random phenomena appear naturally in several branches of science, it is necessary to have systematic tools to formalize and study them. Although probability theory has its roots in efforts to analyze games of chance, today it is the branch of mathematics concerned with analysis of all random phenomena, studying them and formulating laws about their behavior. The book under review is a standard graduate textbook in this area of mathematics that collects various classical and modern topics in a friendly volume.

The book is fluently written, though the explanations of the concepts and theorems are terse. Thus, the readers and the instructors who are going to use this book have the opportunity to learn quickly and efficiently. Moreover, the book contains many exercises. It is a very good source for a course in probability theory for advanced undergraduates and first-year graduate students.

The book has a certain number theoretic flavor in a few interesting results, such as in Chapter 2, where the author gives a probabilistic proof of the Euler product formula for the Riemann zeta function. In Chapter 5 he studies the speed of convergence in the strong law of large numbers, and in Chapter 15 he talks about the important topic of characteristic functions and the central limit theorem. These sections mention results that are similar to those in probabilistic number theory.

The book consists of 26 chapters. In the first chapter, the author gives a brief introduction to measure theory, which is required for the whole text. Since measure theory is a linear theory, it is not useful to describe the dependence structure of events and random variables. Hence the author introduces the concepts of independent events and random variables immediately in the second chapter. The third chapter studies probability generating functions, which is a key idea, relating a class of probability values that are of interest to a class of power series that are easy for computations. In the fourth chapter, based on the notions of measure spaces and measurable maps, the author introduces the integral of a measurable map with respect to a general measure, which is a generalization of the Lebesgue integral.

Studying the median, expectation, and variance, which are the most important characteristic quantities of random variables, is the subject of Chapter 5, where the author gives the laws of large numbers and includes a quick look at the concept of entropy and the source coding theorem. Chapter 6 is devoted to a systematic treatment of almost sure convergence, as well as convergence in measure and convergence of integrals, with the concept of uniform integrability as the key for connecting them. Chapter 7 studies the spaces of functions whose p-th power is integrable, several inequalities concerning them, Lebesgue’s decomposition theorem and the Radon-Nikodym theorem. The concepts of conditional probabilities and conditional expectations are the subjects of Chapter 8.

Martingales are one of the most important concepts of modern probability theory, formalizing the notion of a fair game. The author studies them and related topics in Chapters 9–12. Chapter 13 studies the convergence of measures and Chapter 14 studies probability measures on product spaces. The important topic of characteristic functions and the central limit theorem are studied in Chapter 15, and in Chapter 16 the author studies infinitely divisible distributions.

The author studies Markov chains, their convergence, and their applications in studying electrical networks, respectively in Chapters 17, 18, and 19. Ergodic theory and Brownian motion are the subjects of Chapters 20 and 21, and the law of the iterated logarithm for the Brownian motion is studied in Chapter 22. The concepts of large deviations and the Poisson point theorem are the subjects of Chapters 23 and 24. The author ends the book by studying the Itô integral and stochastic differential equations, in Chapters 25 and 26, which can be useful for the people working on financial mathematics.

Considering the above remarks, and also considering the variety of studied topics, the book should be useful for a wide range of audiences, including students, instructors, and researchers from all branches of science who are dealing with random phenomena.

See also our review of the first edition.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.