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David Williams

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

by David Williams (Author)

4.5 52 ratings

Part of: Cambridge Mathematical Textbooks (21 books)

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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.

ISBN-10

0521406056
ISBN-13

978-0521406055
Publisher

Cambridge University Press
Publication date

14 Feb. 1991
Part of series

Cambridge Mathematical Textbooks
Language

English
Dimensions

15.88 x 1.91 x 23.5 cm
Print length

265 pages
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Product description

Review

'… one of the best introductions to Martingale theory.' Monatshefte für Mathematik

Book Description

This is a masterly introduction to the modern, and rigorous, theory of probability. The author emphasises martingales and develops all the necessary measure theory.

Product details

Publisher ‏ : ‎ Cambridge University Press (14 Feb. 1991)
Language ‏ : ‎ English
Paperback ‏ : ‎ 265 pages
ISBN-10 ‏ : ‎ 0521406056
ISBN-13 ‏ : ‎ 978-0521406055
Dimensions ‏ : ‎ 15.88 x 1.91 x 23.5 cm

Best Sellers Rank: 268,797 in Books (See Top 100 in Books)
- 31 in Differential Equations (Books)
- 341 in Mathematical Probability & Statistics
- 604 in Higher Mathematical Education

Customer reviews:
4.5 52 ratings

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David Williams

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Customer reviews

4.5 out of 5

52 global ratings

5 star		76%
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Sanbi

Delivered in time and in perfect condition

Reviewed in the United Kingdom on 7 April 2021

Verified Purchase

I consider the fact that this classic book is a valuable tool for learning probability and to hone the intuition about probability and martingales a given. So I mainly focus on the delivery and condition of the book which were excellent.

Reviewed in the United Kingdom on 15 February 2018

Verified 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) introduction to martingales.
Based on my own experience, this is an excellent introduction
for anyone starting on probability and martingale theory.

I simply can't recommend this book enough.

Reviewed in the United Kingdom on 28 October 2020

Verified Purchase

Book is great no doubt. The book had some damage and the seller solved the problem right away. Many thanks

Reviewed in the United Kingdom on 24 December 2020

Verified Purchase

Nice book. Nice seller.

Reviewed in the United Kingdom on 5 April 2022

A solid quick introduction to basic measure theoretic probability theory

Reviewed in the United Kingdom on 7 November 2015

nice book

Finally...theory with some perspective....

Reviewed in the United Kingdom on 12 January 2011

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.

Reviewed in the United Kingdom on 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. 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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