Foundations of Modern Probability

"... Indeed the monograph has the potential to become a (possibly even ``the'') major reference book on large parts of probability theory for the next decade or more." M. Scheutzow, Zentralblatt

"The theory of probability has grown exponentially during the second half of the twentieth century and the idea of writing a single volume that could serve as a general reference for much of the modern theory seems almost foolhardy. Yet this is precisely what Professor Kallenberg has attempted in the volume under review and he has accomplished it brilliantly. ... It is astonishing that a single volume of just over five hundred pages could contain so much material presented with complete rigor and still be at least formally self-contained..." R.K. Getoor, Metrica

This new edition contains four new chapters as well as numerous improvements throughout the text.

Olav Kallenberg was educated in Sweden where he received his Ph.D. in 1972 from Chalmers University. After teaching for many years at Swedish universities, he moved in 1985 to the U.S., where he is currently a Professor of Mathematics at Auburn University. He is known for his book Random Measures (4th Edition, 1986) and for numerous research papers in all areas of probability theory. In 1977, he was the second recipient ever of the prestigious Rollo Davidson Prize from Cambridge University. In 1991-1994, he served as the Editor-In-Chief of "Probability Theory and Related Fields."

Kallenberg in his usual rigorous style presents the basic measure theory in the first two chapters. He then covers most of the standard probability theory in the next three chapters. Random variables and processes are covered in chapter 3 with the concepts of convergence and independents and the important zero-one laws. Probability distributions, expectations and higher order moments are also covered in chapter 3.

Chapter 4 deals with random sequences and series and averages and includes the strong law of large numbers and Kolmogorov's three series theorem. Chapter 5 covers characteristic functions and important limit theorems including the central limit theorem (Lindeberg-Feller version).

Conditioning and coupling are covered in Chapter 6 and martingales, submartingales and optional stopping are also covered. Upcrossing inequalities and maxima are also discussed here.

Stochastic processes are covered in chapters 8 - 10 and point processes in chapters 11 and 12. Chapter 13 introduces Gaussian processes and Brownian motion. The law of the iterated logarithm is presented in chapter 13 also. Chapter 14 deals with the important Skorohod embedding technique and invariance principles.

The remaining 13 chapters cover many advanced ideas including convergence of random processes and measures, stochastic integrals and Ito calculus, Feller processes and semi-groups, ergodic theory for Markov processes, stochastic differential equations, diffusions, semi-martingales, large deviations, connections with partial differential equations and more.

This book contains every topic I have seen in texts on advanced probability and more! Kallenberg tends to be both rigorous and elegant in his presentation.

This book is for graduate student and probabilists and mathematical statisticians who need these tools to establish limit theorems. It is not intended for an undergraduate course in probability for non-mathematicians. It requires an understanding of advanced mathematics.


Serious and the Ultimate   January 21, 2002
Dave D (York, PA United States)
11 out of 11 found this review helpful

This is the ultimate graduate textbook in probability. Said that it is important to notice this is not written for the people that have not had a senior level advanced probability course as a matter of fact few such courses.
It is hard to read for the proofs are lightning and lack all the details while the statements are as general and abstract as possible. It can put a serious strain on gaining intuitive understanding. Nevertheless I cannot imagine book any more comprehensive and significant than this one. Using this as a textbook on graduate level will require major input of the instructor and a serious effort for the students.
One bad thing about the book is very dense typesetting that makes it not a very friendly reading.


Incredible book, almost self-contained!   January 27, 2006
a reader (Taipei)
8 out of 8 found this review helpful

This book has accomplished something few people will consider feasible. It covers numerous important topics in 600 pages which ordinarily will be available only in several books and monographs combined together with total number of pages easily exceeding 2000! Nevertheless, the author has done it not by cheating or handwaving as found in some other books - like those familiar words "this topic is beyond the scope of the present book " or "the proof is omitted for lack of space" etc. As far as I have read, the book is in principle self contained. I venture to bet that a reader comfortable with rigorous calculus and rudimentary topology can already attempt tackling it.

But this is not to say that the book is easy. The first two chapters already constitute a reasonable exposition of measure theory and Lebesgue integration, in only 44 pages. The first ten chapters in 200 pages will suffice for a usual graduate course in probability. To accomplish this feat the author has obviously done a lot of soul searching which manifests itself in almost every page. In the author's own words: "I had to be extremely selective in the choice of material...it was necessary to look for the most economical approach to every result I did decide to include... I was surprised to see how much could actually be done to simplify and streamline proofs, often handed down through generations of textbook writers... Over the years I have developed a personal approach to almost every topic, and even experts might find something of interest. " And indeed he has successfully isolated the bare essences of proofs without being unreasonably terse or leaving the reader disorientated - the few misprints and rarely undefined symbols will not be a serious problem for the active reader, although a passive spectator will have a hard time to get through the book's dense forest. Kallenberg's fine balance between brevity and readability will be a delight for mature mathematicians and provide an enormously fruitful exercise for the students.

All in all, Kallenberg has accomplish an amazing survey of probability theory up to the seventies. What about the more recent results? In his 2001 preface, he says that "probability theory keeps growing at an ever increasing rate, and I am painfully aware that I will never catch up in my efforts to survey the entire subject... a comprehensive treatment of the more recent developments would require another volume or two. I am asking for the reader's patience and understanding." The very idea of a comprehensive treatment of contemporary probability in a single volume or two seems quixotic - if it comes from any other person. But Kallenberg has already performed a Mission Impossible. Maybe he will astound us twice in the future!



The BEST book on modern probability   May 24, 2003
7 out of 7 found this review helpful

a) Well organized!
b) Covers a broad range of topics e.g. measure theory, stochastic processes, martingales, Markov processes, stochastic calculus, SDE...and much more!! You can find in it almost all branches of probability!
c) Proofs are short, efficient and interesting, but you have to fill in many details. This gives you a good training!!
d) Results are usually stated in the most general form
e) It requires a strong backgroung in real analysis and functional analysis
e) Very very few typos!