Probability with Martingales (Cambridge Mathematical Textbooks) 1st Edition by David Williams (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I had somehow studied the topic(s) during university and read that book a few years later, working as an engineer. I am not a professional mathematician.This book was quite a thrill, it really tied the pieces together and I believe I have a much deeper understanding after reading it. The lively aspect, I would say, manifests itself in that the examples chosen really drum up interest and the conciseness allows one to have an overview.It’s fairly accessible, proofs included – and even if not, any hole can easily be bridged with the help of stack/mathoverflow nowadays. Some exercises are hard though, but then again it’s alright, it’s not the book on continuous processes from the same author!

⭐This is a fantastic introduction to probability theory great for self-study. The primary focus of the book is discrete time martingales as is obvious from its title, and so it isn’t a complete one semester course (topics which don’t make a mention are infinitely divisible laws, ergodic theory, markov chains, convergence in probability measures to name a few), but neither does it aim to be. The best reason to study this book is the elegant and lively writing style of Williams.

⭐The material in this book is useful, the examples and exercises are great, but I give this only 3 stars because it is badly let down by the proofs. Oftentimes the proofs are stated as being too ‘obvious’, ‘immediate’ to even be written down.Consider Section 6.5, ‘Sum of non-negative random variables’. Out of the 4 (fundamental) statements made in there, 3 are ‘proved’ by merely stating that they are ‘obvious’,’immediate’,’evident’ etcNow this is certainly the case for the author, but the rest of us are buying this book to understand the material, and this attitude borders on being disrespectful.Again, one needs to buy this book for the examples inside, some of them famous , eg the ‘ABRACADABRA’ question, but to actually learn the material and have proper proofs – there are MUCH better books out there, covering the SAME material and with COMPLETE proofs: consider ‘Probability Essentials’ by Jacod for instance. Don’t believe me ? Compare and contrast the proof of the Holder inequality for instance !

⭐Comprehensive and lively. Certainly not a book to begin self-study of probability theory but an excellent reference with plenty of pointers towards deeper applications of martingale theory in continuous time, stochastic integration, and filtering/control applications. The writing style can be terse but clearly motivates each concept.

⭐This is “the book” for Martingale Theory. It is pleasurable to read, and it is one of my favorite texts.

⭐packaging was well done. product description is exactly what it is.

⭐DO NOT BUY THE KINDLE VERSION……many of the equations are not readable……..

⭐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!

⭐This book is extraordinary, beautiful and clear exposition, thoughtfully chosen exercieswhich increase the understand of the topic instead of just moving the reader through the list on autopilot (which manytextbooks 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 introductionfor anyone starting on probability and martingale theory.I simply can’t recommend this book enough.

⭐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.

⭐Nice book. Nice seller.

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

⭐nice book

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