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Download Probability Theory: A Comprehensive Course epub
ISBN: 1848000480
ISBN13: 978-1848000483
Category: No category
Language: English
Publisher: Springer
ePUB size: 1635 kb
FB2 size: 1513 kb
Rating: 4.4
Votes: 515
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Before meeting this superb book, I had read many books on measure-theoretic probability theory likes Billingsley, Athreya and Lahiri, and William (Probability with Martingale). I think that all of them always have both pros and cons, and, IMHO, they deserve 4 stars.

Until I read the paper on D_1 metric for the space of copulas authored by Trutschnig (2011). He cited two probabilty textbooks since he defined D_1 from the generalized version of conditional density called as "(a version of) regular conditional distribution". The first one is the elegant monograph of Kallenberg, which I think it should be the reference on the desk of probabilty theorists or the books for advanced class in graduate level probability theory since the treatment is extremely dense. The second one is this textbook.

As the title said, this textbook is "comprehensive' literally. The first 15 chapters can be considered as a concrete first course in the probability theory for graduate students. It starts from the background on measure theory to nice treatments on martingale, probability on product space (Kolmogorov's extension theorem), and properties of characteristic functions. Although the setup of statements in this part is always in the most generalized version, the ideas and proofs are written in the lucid manner. For me, this text book is self-contained for students with moderately hard effort.

The rest of the book is devoted to the introductory stochastic processes theory. It covers basic treatments such as Markov processes, Brownian motion, Poisson processes, Ito calculus and SDE (crash course), and even somewhat non-standard topics for general probability theory books llikes large deviation and ergodic theory. For me, this part are a nice tour on these topics. After reading it, one can expect the reader to be ready for the specific textbooks on each topic.For example, even when I read Stochastic calculus and SDE such as Oksendal's SDE and Karatzas and Shreve's Brownian motion and stochastic calculus. I consult Klenke's book in terms of both basic background in probability and reference for basic treatment in stochastic processes.

For me, I understand that this book may be dense for the first time, but your effort on this book is worthwhile. If someone looks for an even more gentle "introduction" for non-specialists in probability theory and related fields, I suggest William's Probability with Martingales. For one who wishes to go to the deeper side of probability theory, this niche work is created for you. As a guy graduated with thesis on copula-based dependence measure theory and someone reading SDE himself for preparing himself to get job, I can recommend this book to anyone with mediocre background likes me.
I've only skimmed my library's copy, and so decided to buy it. While aimed at an undergrad level, it appears to be remarkably comprehensive. Although I was already familiar with most areas covered in the book, I'd only picked up my knowledge piecemeal, from many different, and disparate sources. Thus, I was pleasantly surprised to find all this info in just one place.

The book may strike some readers as "dense", and it does resemble a reference work. It is stuffed with precise definitions, reviewing all of the variants of a given definition, carefully making subtle distinctions. This may make it tough slogging for the beginner, but seems ideal for anyone who is already familiar with the subject, and wants to dig a little deeper, or to have a handy introductory reference.
This book actually delivers what it promises. Extremely comprehensive and modern. Superb index for quick reference. Nothing is really left out. I really like the discussion of measure theory and stochastic processes. It really "rounds out" the coverage of probability theory and show's its breadth. Most references omit one or both of these areas. Also, he mercifully includes a table of notation at the back, which he uses consistently throughout all 600+ pages, great for when you jump into a particular result and are hit with a block of symbols! All theorems are proved, which is very nice!
Excellent book. I wrote the author with a minor correction, and he responded very quickly and we figured it out. That is much appreciated! I wish the intro chapters went into a bit more depth and that some of the proofs and examples contained a bit more detail. But this book is a fairly good balance at showing just enough detail.
For a while I was trying to learn Probability Theory from Patrick Billingsley's textbook, which I believe is a standard in many universities (at least in our university that was the recommended reference). Until I was very lucky to stumble upon a copy of this book, very new release (2008 for the English version) at a book fair. I skimmed through it, looked for an Amazon review (there was only one at that time) and was easily convinced, from first impressions, so I bought the single copy at the book fair.

Now that I have read most of it, I think this book stands a good chance to become THE new, stand-alone, standard reference for probability theory (I would vote for it, if there was a poll). It has many wonderful qualities: very clear presentation - I thought the contents could have been ready-made for use as lecture notes, having a wonderful clarity. Quite comprehensive, many proofs are provided, which is perfect for self-learning, and the author gives sufficient hints whenever proofs are shortened.

I appreciate the sections on more fundamental topics such as measure theory, which to me was something new, and very useful. I like how the "pre-requisite" sections were clearly indicated but not all lumped in the beginning. For example, the review of topology, in anticipation of the section on product measure. Very compact, yet rigorous, treatment of regular conditional densities, the Radon-Nikodym derivative, the martingale theory, optional sampling etc. an introduction to stochastic calculus. Topics from applications also abound, and a couple of nice, anectdotal commentaries here and there.

The whole text strikes me as very "modern" and very very clear. To be honest, this has been my favorite book this past year.

If you are a graduate student who wishes to specialize in Probability, this may be a great starting point, to cover a rigorous study of the fundamentals. It does not cover some of the more advanced topics in detail (such as in Protter, Stochastic Integration). If on the other hand you are in a "hurry" to learn probability, perhaps other, shorter books maybe better for you, like Jacod and Protter's Probability Essentials, or Rosenthal's A First Look at Rigorous Probability Theory, both of which are also clear and excellent texts though not nearly as comprehensive like Klenke's.
This is the new best introductory book on probability theory, without a doubt.