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Probability Theory (Universitext), by Achim Klenke
Download PDF Probability Theory (Universitext), by Achim Klenke
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Aimed primarily at graduate students and researchers, this text is a comprehensive course in modern probability theory and its measure-theoretical foundations. It covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as: limit theorems for sums of random variables; martingales; percolation; Markov chains and electrical networks; construction of stochastic processes; Poisson point processes and infinite divisibility; large deviation principles and statistical physics; Brownian motion; and stochastic integral and stochastic differential equations. The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.
- Sales Rank: #1327420 in eBooks
- Published on: 2013-04-11
- Released on: 2013-04-11
- Format: Kindle eBook
Review
From the reviews:
"The book is indeed comprehensive, consisting of 26 chapters on different topics. … can be well used as a reference book on a wide range of topics. The target audience is researchers and graduate students … . Numerous advanced topics are included, so that the book is more inclusive … . There is more than enough material for a two-semester course here. … the book will primarily be used as a reference book. For that purpose, it is a rich and relatively inexpensive choice." (Miklós Bóna, MathDL, January, 2008)
"This book of over 600 pages gives a self-contained presentation of modern probability theory. It is based on courses on advanced probability given by the author. … Most of the proofs are well detailed. … This book will be helpful for graduate students in mathematics … and for researchers in mathematics or theoretical physics." (Sophie Lemaire, Mathematical Reviews, Issue 2009 f)
From the Back Cover
Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.
This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:
- limit theorems for sums of random variables;
- martingales;
- percolation;
- Markov chains and electrical networks;
- construction of stochastic processes;
- Poisson point processes and infinite divisibility;
- large deviation principles and statistical physics;
- Brownian motion; and
- stochastic integral and stochastic differential equations.
The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.
Most helpful customer reviews
26 of 27 people found the following review helpful.
Amazingly well-done
By Christopher Grant
I taught a course in advanced probability out of the first half of Klenke's _Probability Theory_ during Fall 2009 at Brigham Young University, and I'm just starting to teach the follow-on course out of the second half. I am, therefore, thoroughly familiar with the first half of the book but admittedly only vaguely familiar with the second half. I've decided to go ahead and write a review now based on this incomplete information in order to help faculty who may be selecting a probability textbook now for the coming academic year.
In my opinion, this is an extraordinarily good textbook! I've taught classes out of some great books before (e.g., Rudin's _Real and Complex Analysis_, Jones' _Lebesgue Integration on Euclidean Space_, Abbott's _Understanding Analysis_) but I can't remember ever being as impressed with a textbook as I am with Klenke's. His logical arguments are amazingly precise and clear. Even little things like his choices of notation and fonts seem ideal. I think German is Klenke's native language, but his use of English in this book is not stilted at all. The book is mainly self-contained and, in particular, does measure theory from scratch. It was quite a revelation to me to see how clearly and concisely one could work up to Caratheodory's measure extension theorem.
Judging by the copies that appeared on the shelf of the campus bookstore this semester, Springer has not yet subjected Klenke's book to the print-on-demand treatment, so the printing is still nice and sharp. From the perspective of a mathematician and a book lover, _Probability Theory_ is a work of art, and it's been a genuine privilege to get to use it.
18 of 18 people found the following review helpful.
This could be a new Standard Reference for Probability Theory!
By Juan Miguel Montes
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.
15 of 16 people found the following review helpful.
Comprehensive introductory reference
By Linas Vepstas
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.
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