Aimed primarily at graduate students and researchers, this text is a comprehensive course in modern probability theory and its measure-theoretical foundations. It covers a wide variety of topics, many of which are not usually found in introductory textbooks. The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.

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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. 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— 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 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 The concepts of large deviations and the Poisson point theorem are the subjects of Chapters 23 and 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. See the table of contents in pdf format. Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R. Ford Awards Merten M. Achim Klenke. Publication Date:. Number of Pages:. Probability Theory. Log in to post comments.

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## Probability Theory : A Comprehensive Course

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## ISBN 13: 9781848006928

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.

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## Probability Theory: A Comprehensive Course

The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in probability theory. This second edition has been carefully extended and includes many new features. It contains updated figures over 50 , computer simulations and some difficult proofs have been made more accessible. A wealth of examples and more than exercises as well as biographic details of key mathematicians support and enliven the presentation. It will be of use to students and researchers in mathematics and statistics in physics, computer science, economics and biology. Probability Theory : A Comprehensive Course.

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## Probability Theory

It seems that you're in Germany. We have a dedicated site for Germany. The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in probability theory. This second edition has been carefully extended and includes many new features. It contains updated figures over 50 , computer simulations and some difficult proofs have been made more accessible. A wealth of examples and more than exercises as well as biographic details of key mathematicians support and enliven the presentation. It will be of use to students and researchers in mathematics and statistics in physics, computer science, economics and biology.