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Probability with Martingales (Anglais) Broché – 14 février 1991

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Descriptions du produit

Revue de presse

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

Présentation de l'éditeur

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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Format: Broché
Il s'agit vraiment d'un excellent livre. David Williams redémontre d'abord la plupart des résultats de base de théorie de la mesure (tribu produit, théorème de Lebesgue...). Ca n'est peut être pas le premier livre à lire pour qui n'a jamais été initiée à l'intégration de Lebesgue auparavant, mais il permet en tout cas de se rafraichir la mémoire et mieux que ça, d'éclaircir ses idées sur le sujet. L'auteur entre ensuite dans le vif du sujet : les martingales (à temps discret). Les preuves vont toujours droit au but et sont d'une efficacité remarquable. Plusieurs exemples d'applications sont présentés. Un certain nombre de résultats, par exemple sur les fonctions caractéristiques ou les différents modes de convergence - en proba, p.p., sont laissés en annexe mais la plupart du temps démontrés. L'auteur préfère toujours donner les résultats les plus essentiels que d'embrouiller le lecteur avec une multitude de propositions qui tournent autour du même sujet. Le lecteur acquérira des bases solides, et après avoir lu ce livre avec plaisir aura presque sûrement envie d'aller plus loin. Il ne sera cependant peut être pas facile de trouver un livre de niveau plus élevé qui ait de telles qualités pédagogiques.
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Par Un client le 3 février 2003
Format: Broché
This book, which is essentially the set of lecture notes for a third-year undergraduate course at Cambridge University, is a nice textbook on measure-theoretic probability theory.
It is a rigorous and self-contained textbook. After some intuitive introduction, it defines most probability concepts (event, probability, random variable, expectation, etc.) in a rigorous measure-theoretic language, it states many useful key results from measure theory in the main text, and gives complete proofs of these results in appendices.
This book is a modern textbook. It 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.
This book also contains many lively examples and interesting applications. For the applications of martingale theory, it discusses the discrete Black-Scholes formula, the Kalman-Bucy filter, etc.
Another feature is the arrangement of exercises. According to the author, "The most important chapter in this book is Chapter E: Exercises", which consists of the homework sheets given by the author to his students. The author attempts to train the creativeness of his readers; he likes readers to first read the statement of a result, and then to try to prove it for themselves before they read the proof given in the appendices.
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Format: Broché Achat vérifié
No disappointment from this book even if an appendix with the solutions of the exercises would have been a plus

Very deep in the probability, crystal clear, good tips
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Commentaires client les plus utiles sur Amazon.com (beta)

Amazon.com: 4.2 étoiles sur 5 21 commentaires
35 internautes sur 35 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 excellent probability text 24 janvier 2008
Par Michael R. Chernick - Publié sur Amazon.com
Format: Broché
This is an excellently written text on probability theory that emphasizes the martingale approach. The treatment is softer than Neveu's "Discrete Parameter Martingales". Williams intends this book for third year undergraduates with good mathematical training as well as for graduate students.
It provides all the classic results including the Strong Law of Large Numbers and the Three-Series Theorem using martingale techniques for the proofs. It includes many exercises that the author encourages the reader to go through. The author recommends the texts of Billingsley, Chow and Teicher, Chung, Kingman and Taylor, Laha and Rohatgi and Neveu's 1965 probability theory book for a more thorough treatment of the theory.

Measure theory is at the heart of probability and Williams does not avoid it. Rather he embraces it and views probability as both a source of application for measure theory and a subject that enriches it. He covers the necessary measure theoretic groundwork.

However, advanced courses in probability that require measure theory are usually easier to grasp if the student has had a previous mathematics course in measure theory. In the United States, this usually doesn't occur until the fourth year and measure theory is mostly taken by undergraduate mathematics majors. Sometimes it is taken by first year graduate students concurrent with or prior to a course in advanced probability. For these reasons I would advise most instructors to consider it mainly for a graduate course in probability for math or statistics majors.

In the Preface, the author is quick to point out that probability is a subtle subject and honing one's intuition can be very important. He refers to Aldous' 1989 book as a source to help that process. I was disappointed that he didn't mention the two volumes on probability by Feller. Feller's books, particularly volume 2 with his treatment of the waiting time paradox, Benford's law and other puzzling problems in probability is a most stimulating source for appreciating the subtleties of probability, for honing one's intuition and for craving to learn more. It is a shame that Williams didn't mention it there. At some point Williams does refer to Feller's work but he only references volume 1.
51 internautes sur 56 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 For the Probabilist who wants to travel light 10 février 1998
Par Giuseppe A. Paleologo - Publié sur Amazon.com
Format: Broché
This textbook is an introduction to the measure-theoretic theory of probability. The style is unconventional. There is humor here, together with hints and suggestions for the "working probabilist". The first part of the book is rather conventional and introduces the concepts of probability spaces, events, expectation, independence of events. The second part introduces discrete-parameter martingales. Many results are given a "martingale proof". Usually, proofs are elegant and concise (at the cost of not being super-rigorous). For example, existence of conditional expectation is proved using ortogonal projection in L^2 (very nice!). Exercises are interesting and mixed with the text. There are no typos, and the cost of the book is reasonable. I would advise my grandma to buy this book (if she were interested in probability).
15 internautes sur 16 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 A nice treatment of Discreet Time Martingales 24 janvier 2006
Par Abdullah Alothman - Publié sur Amazon.com
Format: Broché
Please Note:I gave it 5 stars because of Chapters 9 - 14.

This book consists of three parts. I will review each in turn:

Part 1: The First 8 Chapters, these covers basic measure theory:

The coverage here is streamlined and the pace is fast. I learnt measure theory as an undergraduate in the mid 80's using the text Measure Theory, also the first 8 chapters, by Halmos. The treatment here, if one is to take appendices 1-3 seriously, is almost at the level of Halmos, but the style, which is geared towards the probabilist, a lot more enjoyable. My only complaint, treatment of product measures and Fubini's theorem in section 8. One would do well to supplement this with the relevant section from Bilingsley's Probability and Measure.

Part2(The Core): Six Chapters on conditional expectation and discreet time martingale theory, one on applications:

The real beauty of this book is - modulu the chapter on applications, which like part three should have been left out - is in chapter 9 -14.

Chapter 9: Is a very nice treatment of conditional expectation. Its existence is proved using basic Hilbert Space Theory rather than the traditional Radon Nikodym - which the author does not develop in part 1-approach. Basic rules for its manipulation are then listed and proven. Armed with this, and the results from part 8 the reader is now finally ready to study martingale theory which is the subject of the next 5 Chapters.

Chapter 10: Is concerned with definitions. Martingales, Submartingales, and Super martingales - collectively called Smartingales, Chung's terminology - are defined. Optional times are defined. A very simple proof showing that Stopped Smartingales are Smartinglaes is given. Various versions of the Optional Sampling theorem - though not the most general, since uniform integrability and hence closure has not yet been defined -are proved.

Chapter 11: Only three pages. This motivates and gives a lovely proof of the Submartingale Upcrossing Theorem. The proof is so intuitive and simple, in marked contrast to that given in Billingsley. Various limit theorems - assuming L1 boundedness - are then proven, though none showing convergence in mean, for these the reader must wait till chapters 12 and 13.

Chapter 12: Defines the concept of L2 bounded martingales. Then digresses for 8 pages. This digression builds some machinery and uses it to prove both Kolmogorovs Three Series Theorem and the Strong Law of Large Numbers. The chapter ends by proving that every process X - in L1 - can be decomposed into:

X = X0 + M + A

Where A is Predicitable null at 0 and M is a Martingale null at 0.

In the case where X is a Submartingale A is shown to be increasing. This is the discreet version of the Doob Meyer decomposition, which says that every cadlag Submartingale is a Semimartingale.

Chapter 13: This introduces the concept of Uniform Integrability.

Chapter 14: Finally we are ready, armed with the machinery developed in 13, to prove convergence results for Uniformly Integrable Martingales. Convergence in L1, Levy's Upward / Downward theorems, Doobs SubMartingale Inequalities are all proven. Finally, a beautiful proof of the Radon Nikodym theorem is provided.

Chapter 15: Applications.

Part3: Three brief chapters dealing with, Characteristic Functions, Weak Convergence, the Central Limit Theorem, in that order. This part, consisting of 20 pages, would have been better left out. It is only the briefest of introductions to these areas, and therefore, given that this is a book on mathematics, should be left out. Instead, I refer the reader to Chapter 5, sections 25, 26 and 27, in Billingsley, for an excellent treatment of the above topics.
22 internautes sur 25 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 eccentric, but wonderful 21 mai 2003
Par Doug R - Publié sur Amazon.com
Format: Broché
The reviewer who rated this a single star gives a decent imitation of Williams' prose style. What he doesn't mention is Williams' infectious enthusiasm for probability, the beautiful proofs, and the conciseness of this book. You should, of course, read Feller vol. 1 first, but this would be my next choice. I'd never really appreciated rigorous probability before reading this book. He shows that it's not all technicalities.
6 internautes sur 6 ont trouvé ce commentaire utile 
4.0 étoiles sur 5 I bought the book after reading it 28 octobre 2009
Par a reader - Publié sur Amazon.com
Format: Broché
I have taught probability to undergraduates (in the US) from Sheldon Ross' A First Course in Probability, which I recommend to any student. I have also read Feller (superb) and Billingsley (superb) for myself, so you may imagine that I am not new to these topics. Learning from Ross is like learning calculus, learning from Billingsley is like learning mathematical analysis. One must progress from one to the other.

On the other hand, even after learning the subject, one is always looking for something concise, consistently engaging, that gives a good view of the subject, allows you to make new connections, and gives you new ideas. Williams' book is all of that. It is not a book to have on a first exposure to the subject, maybe not for a second exposure either -- that will very much depend on what kind of student you are, and what you want to learn, and how you want to learn it. Only some very special students will go unaided through Williams' book on a first reading. But if you have some experience with the subject already (or with measure theory), and you want to broaden your horizons, then this book will allow you to do that. Williams' enthusiasm shines through every page, which is a plus. At this stage in my understanding of the subject, I actually appreciate that the book doesn't go into every detail, but shows more than enough to be a good guide. I didn't give it 5 stars because, to my taste, it should contain more exercises.

Having said all of that (about the book not being suitable for a first reading), I will take it all back, if you have the "correct" intructor teaching you the material: someone who will fill in some gaps when you need it, give you extra exercises, and in general give you that confidence that you need to feel that you are doing the right thing, and not just lost in the woods.

Enjoy!
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