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Rating: 
Summary: Fantastic
Review: A fantastic book for those who wish to gain a comprehensive exposure to rigourous probability theory. Ideal for self-study. Beautiful presentation, with little typos here and there (previous edition). Terrific example of clarity and precision in exposition. All fundamental results. Theory. If you want to spend the due time, and read line by line this book is invaluable and totally tractable. Prerequisite: An elementary course in Analysis (and may be not even that). Bon-appetit.
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Summary: Wow!
Review: 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.
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Summary: Great book
Review: As a Ph.D student in statistics, I found finally a book that relates so many topics in probability at once...
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Summary: Yeah, absolutely excellent
Review: I started reading this book (first edition) during my early graduate years and I still enjoy this book today, now in its second edition and substantially new materials added. This is a masterpiece textbook on probability, at the graduate level, representing the jewels of probability theory and the always deep and fine Russian school. Probability is a mathematical theory for randomeness or chaos. It is well-known nowdays that randomness not only comes from infinite systems but also arises from finite and simple nonlinear systems. An example at the current research front, probability is a necessary tool for studying turbulence, such as Kolmogorov's 1941 theory. Shiyayev is a world-renowned probabilist and has written fundamental books and papers in many areas, such as inference for stochastic process, sequential analysis, etc. This textbook should prove to be a great learning tool and reference for getting into many of the cutting-edge statistical and probability research. As a short note, I find a very useful martingale theorem in this book which I have used in my PhD thesis and later in a paper in AISM. So enjoy reading and you may find some unexpected rewards.
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Summary: The name should be "Basics of the probability"
Review: It covers almost all of the "basic" ideas in probability.This book is one of the most helpful book for me. You will "learn" what is going on in the field of probability theory. The book is especially suitable for self studying. If you want to learn probability theory, buy this book, dig into it, study again and again if you don't understand a particular part. But the book does its best in terms of clarity of exposition, so you can understand with a careful reading. It will provide a strong tool and you can use this tool efficiently in your professional career. But there is a caveat. You might end up with a tool that you will never use again. The book is suitable for professionals and deep minds.
Rating:
Summary: The name should be "Basics of the probability"
Review: It covers almost all of the "basic" ideas in probability. This book is one of the most helpful book for me. You will "learn" what is going on in the field of probability theory. The book is especially suitable for self studying. If you want to learn probability theory, buy this book, dig into it, study again and again if you don't understand a particular part. But the book does its best in terms of clarity of exposition, so you can understand with a careful reading. It will provide a strong tool and you can use this tool efficiently in your professional career. But there is a caveat. You might end up with a tool that you will never use again. The book is suitable for professionals and deep minds.
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Summary: the best in the subject
Review: the book is a must. Written by no one less than one of the the best researchers in probability theory and finance. It is well written, the proofs are clear and the problems proposed are also very helpful. The chapter on martingales provides the best presentation on the issue.
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Summary: Recommended for researchers in probability and statistics
Review: 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.
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Summary: A complete course in probability
Review: 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)
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