Foundations of Modern Probability

Rating:
Summary: The BEST book on modern probability
Review: 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!

Rating:
Summary: Stunning achievement
Review: This a compendium of all the relevant results of probability theory; in the words of the author, a book about "everything". Many reviewers and the author himself have pointed out that this work is similar in bread and depth to Loeve's classical text of the mid 70's. I have never read Neveu, but find this book unique. It is not suited as a textbook, as it lacks the many examples that are needed to absorb the theory at a first pass. It works best as a reference book or a "second pass" textbook: Kallenberg's presentation illustrates new aspects of classical topics such as measure theory, martingales, diffusions, point processes, and covers many advanced topics. The author has been able to pack a large amount of information by carefully organizing the material, and avoiding repetitions. Although rigorous and advanced, the proofs are elegant and clear. The goal is not to break the world record for conciseness (that is currently held by Borkar's booklet on probability). Kallenberg begins each chapter with useful remarks, so that the goals are always evident. There are a few typos here and there, but nothing that cannot be easily spotted (unlike Durrett's Probability). Over time, this has become my prominent reference source (and I am using only the first 15 chapters of the book...).

Rating:
Summary: The BEST book on modern probability
Review: This a compendium of all the relevant results of probability theory; in the words of the author, a book about "everything". Many reviewers and the author himself have pointed out that this work is similar in bread and depth to Loeve's classical text of the mid 70's. I have never read Neveu, but find this book unique. It is not suited as a textbook, as it lacks the many examples that are needed to absorb the theory at a first pass. It works best as a reference book or a "second pass" textbook: Kallenberg's presentation illustrates new aspects of classical topics such as measure theory, martingales, diffusions, point processes, and covers many advanced topics. The author has been able to pack a large amount of information by carefully organizing the material, and avoiding repetitions. Although rigorous and advanced, the proofs are elegant and clear. The goal is not to break the world record for conciseness (that is currently held by Borkar's booklet on probability). Kallenberg begins each chapter with useful remarks, so that the goals are always evident. There are a few typos here and there, but nothing that cannot be easily spotted (unlike Durrett's Probability). Over time, this has become my prominent reference source (and I am using only the first 15 chapters of the book...).

Rating:
Summary: Serious and the Ultimate
Review: 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 senior level advanced probability course.
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. 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 serious effort for the students.

Rating:
Summary: clear and detailed account of probability
Review: When I was a graduate student at Stanford I took a seminar on point processes taught by Ross Leadbetter who was visiting Stanford for the summer. We used Kallenberg's book "Random Measures". That book provided a concise and mathematically rigorous treatment of random measures. This text on probability is a much larger volume but is masterfully presented.

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.

Rating:
Summary: clear and detailed account of probability
Review: When I was a graduate student at Stanford I took a seminar on point processes taught by Ross Leadbetter who was visiting Stanford for the summer. We used Kallenberg's book "Random Measures". That book provided a concise and mathematically rigorous treatment of random measures. This text on probability is a much larger volume but is masterfully presented.

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.