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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.

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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 By using our website you agree to our use of cookies.

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Probability Theory: A Comprehensive Course by Achim Klenke

Vladimir A. Joseph J. Jean Jacod. Achim Klenke. Ferdinand Verhulst.


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A Comprehensive Course

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.


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  • 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.

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    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. Product details Format Paperback pages Dimensions x x