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Probability with Martingales (Cambridge Mathematical Textbooks) [Paperback]

David Williams
4.2 out of 5 stars  See all reviews (10 customer reviews)
RRP: £34.99
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Book Description

14 Feb 1991 0521406056 978-0521406055
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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Product details

  • Paperback: 265 pages
  • Publisher: Cambridge University Press (14 Feb 1991)
  • Language: English
  • ISBN-10: 0521406056
  • ISBN-13: 978-0521406055
  • Product Dimensions: 22.6 x 15.2 x 1.3 cm
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (10 customer reviews)
  • Amazon Bestsellers Rank: 117,293 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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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.

Inside This Book (Learn More)
First Sentence
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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Customer Reviews

Most Helpful Customer Reviews
14 of 14 people found the following review helpful
4.0 out of 5 stars glass is half full 1 May 2006
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.
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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
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
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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5 of 6 people found the following review helpful
5.0 out of 5 stars Excellent introduction to probabilty theory 16 Dec 2002
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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1 of 1 people found the following review helpful
2.0 out of 5 stars Misses the mark for me 29 April 2013
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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