Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
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'… one of the best introductions to Martingale theory.' Monatshefte für Mathematik
Book Description
This key book is a rigorous account which has Doob's theory of martingales in discrete time as its main theme. A distinguishing feature of this study is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals.
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The purpose of this chapter is threefold: to take something which is probably well known to you from books such as the immortal Feller (1957) or Ross (1976), so that you start on familiar ground; to make you start to think about some of the problems involved in making the elementary treatment into rigorous mathematics; and to indicate what new results appear if one applies the somewhat more advanced theory developed in this book. Read the first page
Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
The purpose of this chapter is threefold: to take something which is probably well known to you from books such as the immortal Feller (1957) or Ross (1976), so that you start on familiar ground; to make you start to think about some of the problems involved in making the elementary treatment into rigorous mathematics; and to indicate what new results appear if one applies the somewhat more advanced theory developed in this book. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Most Helpful Customer Reviews
13 of 13 people found the following review helpful
By Ethan Frome
Format: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.
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.
7 of 7 people found the following review helpful
By A Customer
Format: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.
5 of 5 people found the following review helpful
By a student
Format: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.
Any pure mathematician who thinks that anything involving the word 'probability' is non-rigorous hand-waving should read this.
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Most Recent Customer Reviews
2.0 out of 5 stars
Misses the mark for me
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... Read more
Published 22 days ago by Gorgeous Nicko
5.0 out of 5 stars
A must HAVE book
This book is mainly based upon the measure theory & less probability but still it covers almost all major parts of martingales
Published 12 months ago by Riz
5.0 out of 5 stars
Finally...theory with some perspective....
I find this book a refreshing change from the heavily rigorous texts on probability that I never got around to reading. Read more
Published on 12 Jan 2011 by objectivist
4.0 out of 5 stars
Very useful, eventually
I have to agree with the first comment about Chapter 0 being out of place and confusing. Save this until last. Read more
Published on 14 Dec 2009 by steele_books
4.0 out of 5 stars
coincise book
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!
Published on 17 Feb 2009 by Facchetti Giuseppe
5.0 out of 5 stars
A excellent treatment of modern probability
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... Read more
Published on 19 Sep 2002
5.0 out of 5 stars
excellent; a rigorous introduction to modern probability
Somewhat dry - and, intermittently, dryly humorous - but a fantastic introduction to modern probability theory, at around the final-year undergrad level. Read more
Published on 5 Feb 2002 by Ian Martin
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