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Probability with Martingales (Cambridge Mathematical Textbooks) Paperback – 14 Feb 1991

by David Williams (Author)

13 customer reviews

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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: 15.2 x 1.7 x 22.8 cm
Average Customer Review: 13 customer reviews
Amazon Bestsellers Rank: 96,785 in Books (See Top 100 in Books)
- #15 in Books > Science & Nature > Mathematics > Calculus & Mathematical Analysis > Differential Equations
- #150 in Books > Science & Nature > Mathematics > Probability & Statistics
- #270 in Books > Textbooks > Science & Mathematics > Mathematics

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Review

'… 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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Hayk

Excellent book

15 February 2018

Format: PaperbackVerified Purchase

This book is extraordinary, beautiful and clear exposition, thoughtfully chosen exercies
which increase the understand of the topic instead of just moving the reader through the list on autopilot (which many
textbooks unfortunaltely do).

My research is mainly in PDEs and Harmonic Analysis,
and this book was one of my first (proper) introductions to martingales.
Based on my experience, this is an excellent introduction
for anyone starting on probability and martingale theory.

I simply can't recommend this book enough.

Comment

7 November 2015

Format: PaperbackVerified Purchase

nice book

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25 May 2012

Format: PaperbackVerified Purchase

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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Gorgeous Nicko

Misses the mark for me

29 April 2013

Format: 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.

2 people found this helpful

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Ian Martin

excellent; a rigorous introduction to modern probability

5 February 2002

Format: 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.

7 people found this helpful

Comment

1 May 2006

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.

20 people found this helpful

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A customer

A excellent treatment of modern probability

19 September 2002

Format: 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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Most recent customer reviews

Samuel Johnston

This book is fantastic, but highly frustrating

This book is fantastic, but highly frustrating. I've made several attempts to read it over the last two years and have given up several times time, but recently I set myself the...Read more

Published 1 year ago

objectivist

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 January 2011

steele_books

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 December 2009

Facchetti Giuseppe

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 February 2009

a student

Excellent introduction to probabilty theory

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.Read more

Published on 16 December 2002

A customer

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.Read more

Published on 30 April 2001

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