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14 of 14 people found the following review helpful
4.0 out of 5 stars glass is half full
This book has many problems. Chapter 0 is not something that should be read until after one has read the rest of the book. Some of the exercises are incredibly difficult. Indeed, there is an exercise in chapter 1 (i.e. early so the author should be building ones confidence in the book) which I think most professors of probability would struggle with. The exercises at the...
Published on 1 May 2006 by Ethan Frome

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7 of 7 people found the following review helpful
3.0 out of 5 stars A book that should be used, but not by itself
I am a final year undergraduate using this book for one of my stats courses. The book certainly does not lack in detail, but I feel it could be a bit more user friendly. I personally would have liked to see lots more worked examples etc! However, put together with old exams/ worked examples/other sources of practice questions, this book is definitely a very useful study...
Published on 30 April 2001


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14 of 14 people found the following review helpful
4.0 out of 5 stars glass is half full, 1 May 2006
This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
This book has many problems. Chapter 0 is not something that should be read until after one has read the rest of the book. Some of the exercises are incredibly difficult. Indeed, there is an exercise in chapter 1 (i.e. early so the author should be building ones confidence in the book) which I think most professors of probability would struggle with. The exercises at the back of the book range from the similarly intractable to the reasonably useful but most readers will be wasting much of their time if they persist with these exercises. There are of course no solutions to these.

The book is full of annoying "this is now obvious" or "left to the reader as an exercise" statements. Years later on returning to the book I realise that those statements ARE obvious, that is obvious to someone who has spent years in the subject. It has been my experience that when you read an introductory book on a subject, very few things are obvious and "left to the reader as an exercise" leads to difficulties.

Despite this, it is actually a very good book. The proofs are rigorous, though they could do with a little more explanation in parts. The book covers a great many important topics in a relatively short number of pages and will prepare you well for future studies in stochastics for example. Its coverage of martingales is excellent and its treatment of both conditional expectation and ordinary expectation are both good. One testament to its quality is that when reading other books on stochastic processes (say) I am amazed at how often the results in williams are applied. What I mean by this is that Williams has selected the most useful results and packed them into this small book.

Do read this book, but don't expect it to be easy. It states on the back that it is intended for undergraduates but to be fair it is at a slightly higher level than that. If your'e doing a phd in probability/stochastics/math finance then this is a good place to start. Just remember to look for the positives in it.
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7 of 7 people found the following review helpful
3.0 out of 5 stars A book that should be used, but not by itself, 30 April 2001
By A Customer
This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
I am a final year undergraduate using this book for one of my stats courses. The book certainly does not lack in detail, but I feel it could be a bit more user friendly. I personally would have liked to see lots more worked examples etc! However, put together with old exams/ worked examples/other sources of practice questions, this book is definitely a very useful study aid.
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5 of 5 people found the following review helpful
5.0 out of 5 stars excellent; a rigorous introduction to modern probability, 5 Feb 2002
This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
Somewhat dry - and, intermittently, dryly humorous - but a fantastic introduction to modern probability theory, at around the final-year undergrad level. I used it as an accompaniment to a course on probability and measure at Cambridge.
The book is broken into three parts:
1) Foundations (which starts with some measure theory, continues with integration and a proper treatment of random variables, expectation, etc)
2) Martingale theory (the meat of the book - conditional expectation, martingale convergence, L2 and UI martingales; also includes proofs of zero-one law, strong law of large numbers amongst others)
3) Characteristic functions (tacked on at the end in order to include a proof of the central limit theorem).
The treatment throughout is rigorous; but the author's enthusiasm for his subject is refreshingly apparent.
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1 of 1 people found the following review helpful
2.0 out of 5 stars Misses the mark for me, 29 April 2013
This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
I have honestly tried and tried to love this book, but have not yet succeeded. Everything is there and rigorously explained, but somehow the message more often than not seems to come out scrambled. The material is not all that intrinsically difficult in itself, as higher maths goes (In my own order of preference: Ash, Rosenthal, Resnick, Gut and others all render it far more accessible in my experience) but reading Williams always makes me feel that, if only he and I were both on the same wavelength, his book would be fantastic - sadly for me, we aren't.
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5 of 6 people found the following review helpful
5.0 out of 5 stars Excellent introduction to probabilty theory, 16 Dec 2002
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This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
To discuss random variables in a mathematically meaningful way one must use measure theory. This book gives a concise and insightful introduction to measure-theoretic probability. The material is presented in an order and a pace at which maintains the readers interest. This is a 'must-read' for those wanting to understand measure-theoretic probability and rigorous discrete-time stochastic processes. It is set at a final year undergraduate (or perhaps MSc level).
Any pure mathematician who thinks that anything involving the word 'probability' is non-rigorous hand-waving should read this.
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5.0 out of 5 stars Finally...theory with some perspective...., 12 Jan 2011
This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
I find this book a refreshing change from the heavily rigorous texts on probability that I never got around to reading. The author always provides motivations and "look-aheads" for theorems. Though the emphasis of the text is on martinagles, the initial chapters on measure theory are a great way to get acquainted with the basics of the theory of probability.
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4.0 out of 5 stars Very useful, eventually, 14 Dec 2009
This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
I have to agree with the first comment about Chapter 0 being out of place and confusing. Save this until last. The book is rather concise, which was painful at first and a lot of time was spent filling in the gaps. But once the book had been digested it really came into its own as a quick measure theoretic probability reference book. I give it 4 stars instead of 5 because I can't help wondering if there was an easier way to teach the material, which now seems a lot easier 1 year after reading it. It is however a must read for anyone considering a PhD in probability.
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4.0 out of 5 stars coincise book, 17 Feb 2009
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Facchetti Giuseppe "Giu" (Italy) - See all my reviews
(REAL NAME)   
This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
Modern theory of martingale. the book is very coincise and focous on the use of theorems and definitions. Short but essential discussions are present and precious!
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2 of 3 people found the following review helpful
5.0 out of 5 stars A excellent treatment of modern probability, 19 Sep 2002
By A Customer
This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
This book is simply the best introductory book on modern probability available. It treats the subject matter with sufficient rigour and makes sensible omissions of unnecessary proofs. I have used this as a text for a MSc course in probability and its direct and concise approach makes it a very suitable text for this level - though it is not for the feint hearted.
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0 of 1 people found the following review helpful
5.0 out of 5 stars A must HAVE book, 25 May 2012
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This review is from: Probability with Martingales (Cambridge Mathematical Textbooks) (Paperback)
This book is mainly based upon the measure theory & less probability but still it covers almost all major parts of martingales
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Probability with Martingales (Cambridge Mathematical Textbooks)
Probability with Martingales (Cambridge Mathematical Textbooks) by David Williams (Paperback - 14 Feb 1991)
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