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**Probability. Theory and examples.
5th edition.**
*(English)*
Zbl 1440.60001

Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge: Cambridge University Press (ISBN 978-1-108-47368-2/hbk; 978-1-108-58458-6/ebook). xii, 419 p. (2019).

This book contains an intensive discussion on the main topics in modern probability following the measure-theoretic approach: limit theorems such as law of large numbers and central limit theorem, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. In a master style the author introduces the notions, the concepts and for each one presents the most important results. He pays good attention to both the ideas behind the statements and the technical details in the proofs. The author follows a well accepted format by giving the definitions followed by clearly formulated theorems or lemmas and then providing their proofs. After many of the proofs there are useful remarks and challenging examples. It is remarkable to see successful using of counterexamples to illustrate the importance of conditions under which a particular statement is, or is not, true. Well selected exercises are included within or at the end of most of the sections and subsections. Besides complementing well the material, the exercises give a chance to the reader to be an ‘active player in the game’. Not only is the author well familiar with contemporary developments in probability theory, but he is successful in choosing or extracting specific delicate results from original sources and finding an appropriate form to include them into this book as theorems, lemmas, remarks, examples, counterexamples or exercises.

As mentioned by the author, the present 5th edition, compared with the previous 4th edition [Zbl 1202.60001], contains a new chapter on recent developments on using the multidimensional Brownian motion in relation to PDEs (e.g., Feynman-Kac type formulas).

In brief, this book contains a huge amount of excellent material from contemporary probability, the style is compact but the presentation is clear and complete.

A few groups of readers can essentially benefit from this book. Among them are teachers of university graduate courses in probability and/or measure theory, and students attending such courses. Researchers in several areas can use the book as a reference. And …the book is just a source for intellectual pleasure!

Long ago the book has become, steadily stays, and will stay for many years to come, in the so called group of ‘classic books’. There are more than good reasons to congratulate the author for his great work and standing in science, and also Cambridge University Press for publishing this book in the prestigious series ‘Statistical and Probabilistic Mathematics’.

As mentioned by the author, the present 5th edition, compared with the previous 4th edition [Zbl 1202.60001], contains a new chapter on recent developments on using the multidimensional Brownian motion in relation to PDEs (e.g., Feynman-Kac type formulas).

In brief, this book contains a huge amount of excellent material from contemporary probability, the style is compact but the presentation is clear and complete.

A few groups of readers can essentially benefit from this book. Among them are teachers of university graduate courses in probability and/or measure theory, and students attending such courses. Researchers in several areas can use the book as a reference. And …the book is just a source for intellectual pleasure!

Long ago the book has become, steadily stays, and will stay for many years to come, in the so called group of ‘classic books’. There are more than good reasons to congratulate the author for his great work and standing in science, and also Cambridge University Press for publishing this book in the prestigious series ‘Statistical and Probabilistic Mathematics’.

Reviewer: Jordan M. Stoyanov (Sofia)

### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60A05 | Axioms; other general questions in probability |

97K50 | Probability theory (educational aspects) |