Product Description

This book contains a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, ergodic theory, weak convergence of probability measures, stationary stochastic processes, and the Kalman-Bucy filter. Many examples are discussed in detail, and there are a large number of exercises. The book is accessible to advanced undergraduates and can be used as a text for self-study.

This new edition contains substantial revision and updated references. The reader will find a deeper study of topics such as the distance between probability measures, metrization of weak convergence, and contiguity of probability measures. Proofs for a number of some important results which were merely stated in the first edition have been added. The author has included new material on the probability of large deviations, on the central limit theorem for sums of dependent random variables, and on a discrete version of Ito's formula.

Product Details

• Amazon Sales Rank: #180365 in Books
• Published on: 1995-12-08
• Original language: English
• Number of items: 1
• Binding: Hardcover
• 621 pages

Review

About the First English Edition:

It is clear that this book contains important and interesting results obtained through a long time period, beginning with the classical Bernoulli's law of large numbers, and ending with very recent results concerning convergence of martingales and absolute continuity of probability measures. Let us note especially that the great number of ideas, notions and statements in the book are well-motivated, explained in detail and illustrated by suitably chosen examples and a large number of exercises. Thus, the present book is a synthesis of all significant classical ideas and results, and many of the major achievements of modern probability theory. In the whole it is a welcome addition to mathematical literature and can become an indispensable textbook for courses in stochastics.

- J. Stoyanov, Zentralblatt

Language Notes
Text: English (translation)
Original Language: Russian

Customer Reviews

Recommended for researchers in probability and statistics
The most important quality to note about this book is that it is *indeed* a graduate textbook. Thus, anyone who is considering reading this book and who has not yet studied ideas such as random processes (including Markov chains and martingales), the Central Limit Theorem, etc.. should pass on this book and look into one or more undergraduate texts (Ross's "Probability Models" would make a good start). Another question the potential buyer should ask is, "Do I want to apply probability, or do I want to study probability for the sake of probability?". If it is the former, then there are many more suitable books whose union covers all the topics in this book, and does so in a much more clear and inuitive manner (For example, "Ergodic Theory" by Halmos, "Probability Models" and "Random Processes" by Ross, and "Mathematical Statistics Chapters 1-8" by Roussas is one example of such a covering). On the other hand, if the answer is the latter, then stop here and begin reading! In other words, the 5 stars for this review is relative to anticipating that the reader is a grad. student who wants to do research in probability and/or statistics. Conditioning on these assumptions, the clarity and coverage found in this text cannot be matched. The measure-theoretic approach may take some time to get used to, but in the end the reader will be thankful for this, as many of the proofs fall out quite easily.

A complete course in probability
When I started to develop an interest in statistics and probability I took the challenge of finding a concise probability book that balanced both the formal aspects of the theory as the practical approach. After many books, I decided to take a look a springer-verlag's related texts. I was surprised when I found this book since, even though its name (Graduate Texts in Mathematics, me being an undergrad), it is a down-to-earth book that takes by the hand through probability theory. Among the many books I read, (in some cases just tried to read), Shiryaev's book was the most organized and efficient one: it didn't lose valuable time explaining obvious concepts nor did it overshadow the underlying ideas with complex mathematical formalisms: it was simple and comprehensive. I think this book is highly recommendable to any science student familiarized with the most essential mathematical concepts, who wishes to have in the bookshelf and near of hand a complete probability course (in reality, the only knowledge needed is elementary calculus, series, and a set theory and a very basic notion of probability doesn't hurt). This book, in combination with an applied probability book, is more than enough to understand this fascinating and necessary subjet (I also recommend in adition to this text, "Mathematical Statistics" by Terrell, also a Springer-Verlag)

Wow!
Almost everything has been said already by other reviewers, I am just going to confirm it really is that good. Starts from the very basics, builds theory up to startionary random processes, L2 theory, and some basic ergodic theorems, and gets you ready to learn the theory of random processes (which is not what this textbook is about). You'll have to find some other book as your "Part 2", I don't have any recommendations off-hand. However, for basic probability, in a rigorous setting, this is the book. He explains everything, and doesn't skip too many details, which is sweet. Proofs are easy to follow. However, I have found a few nasty typos which may have you bang your head against the wall. But you'll find them if you read carefully. Hopefully they will be corrected in subsequent printings. This shouldn't deter you. It really is an amazing book, in the russian tradition. See also Ya. Sinai's minibook on probability theory, many theorems are proved without the use of measure theory, so you can compare the proofs and get better insight into the machinery of those theorems.