What do you think?
Rate this book
An Introduction to Probability Theory and Its Applications, Volume 2
The classic text for understanding complex statistical probability
An Introduction to Probability Theory and Its Applications offers comprehensive explanations to complex statistical problems. Delving deep into densities and distributions while relating critical formulas, processes and approaches, this rigorous text provides a solid grounding in probability with practice problems throughout. Heavy on application without sacrificing theory, the discussion takes the time to explain difficult topics and how to use them. This new second edition includes new material related to the substitution of probabilistic arguments for combinatorial artifices as well as new sections on branching processes, Markov chains, and the DeMoivre-Laplace theorem.
An Introduction to Probability Theory and Its Applications offers comprehensive explanations to complex statistical problems. Delving deep into densities and distributions while relating critical formulas, processes and approaches, this rigorous text provides a solid grounding in probability with practice problems throughout. Heavy on application without sacrificing theory, the discussion takes the time to explain difficult topics and how to use them. This new second edition includes new material related to the substitution of probabilistic arguments for combinatorial artifices as well as new sections on branching processes, Markov chains, and the DeMoivre-Laplace theorem.
669 pages, Paperback
First published January 1, 1996
5 people are currently reading
271 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 4 of 4 reviews
March 3, 2015
This is the "go to" book on probability theory. I first encountered it over 30 years ago in graduate school. It is a mathematicians book for non-mathematicians like myself. That is, it is fully rigorous but without the "wedding cake" notation and jargon of more recent texts. Its title says it is an introduction and, in many ways it is, but it also covers a number of advanced topics. Its chapter on fluctuations in coin tossing is one of the best written on the subject and gives the student a real feel for what's going on. The book has many, many, compelling examples. The problems, for the most part, are well stated and most readers will find they can be worked out though requiring varying degrees of effort. Even the easy problems are not trivial (in the sense of being silly exercises) and the difficult ones are probably within the reach of most math students (though I make no claim to have worked them all! But have over the decades tackled most of them).
April 7, 2025
After attaining the summit of a masterly style in Vol. I (our review here), perhaps it is too much to expect for Feller to maintain his productivity at the same level as he continues his upper undergraduate-level exposition of probability theory in Vol. II. To be sure, he tackles here rather more advanced material, up to and including stochastic processes from the standpoint of all the newly minted technique that stands at his disposal, and in which he played no small part in developing in the first place. Certainly, the topics addressed here in the present second volume are appropriate and necessary to anyone’s education in probability theory.
In the first three chapters, special probability density functions expressible in analytic form, such as the exponential, uniform and gamma functions, in one or in higher dimension. In order to lay out the new techniques, a modicum of measure theory as applied to probability is indispensable. This subject occupies chapters four and five. But it is no end in itself, the purpose behind introducing it is to go on to non-trivial uses of it in the remainder of the book. Chapter six surveys examples of some important distributions and processes: stable, infinitely divisible, independent increments, ruin and compound Poisson processes, renewal processes, random walks, queuing processes and general Markov chains.
After thus trotting out a fair sampling of the conventional use cases, Feller turns to a study of them invoking rather harder analysis than anything done in Vol. I: laws of large numbers, the central limit theorem for i.i.d random variables, ergodic theorems for Markov chains, infinitely divisible distributions and semi-groups and Markov processes from the semi-group point of view, as well (chapters seven to ten).
Chapter eleven revisits renewal theory and chapter twelve, random walks on the real line. With the aid of more sophisticated techniques, one can derive new results, such as the renewal theorem, existence of a limiting process, the Wiener-Hopf factorization and the arcsine law. The next two chapters (thirteen and fourteen) introduce the Laplace transform, Tauberian theorems and the method of resolvents up to the Hille-Yosida theorem. By the author’s own admission, the presentation tends to be sketchy and is meant mainly for culture, especially the applications in chapter fourteen, leading up to Kolmogorov’s differential equations.
So much for the Laplace transform. The last part of Vol. II (chapters fifteen to nineteen) concerns the Fourier transform. It might come as something of a surprise that the Fourier transform of a random variable would be an object rewarding investigation, but it turns out to be just what is needed to prove the central limit theorem under the most general Lindeberg conditions and Berry-Esséen’s sharp bound on its rate of convergence. In chapter seventeen, the Fourier transform’s nice behavior under convolution proves very helpful to a closer study of infinitely divisible distributions. For instance, one can work out the domains of attraction of a stable distribution. Fourier methods are also well adapted to random walks and stationary stochastic processes, the subject of chapter eighteen resp. nineteen. This reviewer, however, finds these sections to proceed too rapidly and to be too technical for ready comprehension. They deserve a textbook-length treatment in their own right.
After this brief overview of the expansive contents contained in this second volume, let us remark on Feller’s characteristic style. His proofs are clear and stay at about as elementary level as the material permits; he isn’t one to dazzle the reader with a pompous display of technical virtuosity, when a relatively simple approach will suffice. In this reviewer’s opinion, the enterprising student should aim to acquire familiarity with both perspectives, that of the elementary level (such as Feller’s) and that of the advanced level employing all the functional-analytic tools (such as one might find, say, in A.N. Shiryaev’s Probability, Springer Verlag, 1991, reviewed by us here). Why? Obviously, a more concrete picture facilitates understanding on the part of the beginner. But, if one were to halt there, he would miss the chance to progress up to the state of the art. The current state of the art is defined by challenging problems which have yet to yield to solution based on known methods. Hence, there would be little hope of contributing to it if equipped only with a rudimentary preparation. Feller’s present second volume is thus suited to undergraduates but not really to graduate students, except perhaps those who do not intend to become professional mathematicians.
In any case, the present work will serve well as a first introduction. All the more so given the many worked examples, which are always good. Nevertheless, one cannot escape from the impression that Feller’s pedagogical technique is not quite as refined here as it was in the first volume. There are not so many side comments as there were in the first volume, included to help motivate the reader to see where an idea comes from or where it leads. To be sure, the exposition lays everything out for you, all the steps leading to the result. But it remains, so to speak, less well digested. This reviewer, at any rate, finds it rather tedious to wade through reams of fairly technical derivations without at least some pointers as to what to look for in them. These comments apply particularly to the second half. Perhaps this shortcoming is a problem endemic to textbooks at a level more advanced than the beginner’s, close, that is, to the recent research literature. The temptation is merely to spill out as much as one can, whereas, when teaching more introductory courses one typically is forced to exercise selectivity, and will have had the occasion to see how students receive one’s choices of what to present and how. Therefore, a born pedagogue such as Feller does achieve a degree of self-reflection and polish in the first volume that the second volume doesn’t quite match.
Another observation would be that the more difficult material of the second volume was still relatively new, hence less worked-over, at the time of writing. For instance, this reviewer, having already learned real analysis and measure theory from later textbooks such as Royden’s (our review here) or Folland’s (our review here), could not discern anything noteworthy in chapters four and five of the present work. Thus, one would be better off skipping them and focusing on what is unique to probability theory, as for instance, the concepts of infinitely divisible and stable distributions, treated exhaustively here. The same remark would probably apply to Feller’s presentation of the theory of semi-groups of operators in chapters thirteen and fourteen and of harmonic analysis in chapter nineteen.
These topics, we want to suggest, would be better studied in more modern textbooks, to perfect one’s understanding. Thus, what one should concentrate on here would be the applications special to probability theory itself, such as the use of Laplace transforms in chapter fourteen or of characteristic functions in chapter fifteen. If one warns oneself not to be misled by the table of contents to think that everything one needs to know can be found in this second volume, but is judicious about what to pick up here and what to learn elsewhere, it can be a profitable exercise to go through Vol. II. For, as we have indicated, its level is uneven. Section five in chapter one on ‘The persistence of bad luck’ is matchless! If only Feller had not been prevented by his untimely death from bringing the rest of it up to this standard! Three stars.
In the first three chapters, special probability density functions expressible in analytic form, such as the exponential, uniform and gamma functions, in one or in higher dimension. In order to lay out the new techniques, a modicum of measure theory as applied to probability is indispensable. This subject occupies chapters four and five. But it is no end in itself, the purpose behind introducing it is to go on to non-trivial uses of it in the remainder of the book. Chapter six surveys examples of some important distributions and processes: stable, infinitely divisible, independent increments, ruin and compound Poisson processes, renewal processes, random walks, queuing processes and general Markov chains.
After thus trotting out a fair sampling of the conventional use cases, Feller turns to a study of them invoking rather harder analysis than anything done in Vol. I: laws of large numbers, the central limit theorem for i.i.d random variables, ergodic theorems for Markov chains, infinitely divisible distributions and semi-groups and Markov processes from the semi-group point of view, as well (chapters seven to ten).
Chapter eleven revisits renewal theory and chapter twelve, random walks on the real line. With the aid of more sophisticated techniques, one can derive new results, such as the renewal theorem, existence of a limiting process, the Wiener-Hopf factorization and the arcsine law. The next two chapters (thirteen and fourteen) introduce the Laplace transform, Tauberian theorems and the method of resolvents up to the Hille-Yosida theorem. By the author’s own admission, the presentation tends to be sketchy and is meant mainly for culture, especially the applications in chapter fourteen, leading up to Kolmogorov’s differential equations.
So much for the Laplace transform. The last part of Vol. II (chapters fifteen to nineteen) concerns the Fourier transform. It might come as something of a surprise that the Fourier transform of a random variable would be an object rewarding investigation, but it turns out to be just what is needed to prove the central limit theorem under the most general Lindeberg conditions and Berry-Esséen’s sharp bound on its rate of convergence. In chapter seventeen, the Fourier transform’s nice behavior under convolution proves very helpful to a closer study of infinitely divisible distributions. For instance, one can work out the domains of attraction of a stable distribution. Fourier methods are also well adapted to random walks and stationary stochastic processes, the subject of chapter eighteen resp. nineteen. This reviewer, however, finds these sections to proceed too rapidly and to be too technical for ready comprehension. They deserve a textbook-length treatment in their own right.
After this brief overview of the expansive contents contained in this second volume, let us remark on Feller’s characteristic style. His proofs are clear and stay at about as elementary level as the material permits; he isn’t one to dazzle the reader with a pompous display of technical virtuosity, when a relatively simple approach will suffice. In this reviewer’s opinion, the enterprising student should aim to acquire familiarity with both perspectives, that of the elementary level (such as Feller’s) and that of the advanced level employing all the functional-analytic tools (such as one might find, say, in A.N. Shiryaev’s Probability, Springer Verlag, 1991, reviewed by us here). Why? Obviously, a more concrete picture facilitates understanding on the part of the beginner. But, if one were to halt there, he would miss the chance to progress up to the state of the art. The current state of the art is defined by challenging problems which have yet to yield to solution based on known methods. Hence, there would be little hope of contributing to it if equipped only with a rudimentary preparation. Feller’s present second volume is thus suited to undergraduates but not really to graduate students, except perhaps those who do not intend to become professional mathematicians.
In any case, the present work will serve well as a first introduction. All the more so given the many worked examples, which are always good. Nevertheless, one cannot escape from the impression that Feller’s pedagogical technique is not quite as refined here as it was in the first volume. There are not so many side comments as there were in the first volume, included to help motivate the reader to see where an idea comes from or where it leads. To be sure, the exposition lays everything out for you, all the steps leading to the result. But it remains, so to speak, less well digested. This reviewer, at any rate, finds it rather tedious to wade through reams of fairly technical derivations without at least some pointers as to what to look for in them. These comments apply particularly to the second half. Perhaps this shortcoming is a problem endemic to textbooks at a level more advanced than the beginner’s, close, that is, to the recent research literature. The temptation is merely to spill out as much as one can, whereas, when teaching more introductory courses one typically is forced to exercise selectivity, and will have had the occasion to see how students receive one’s choices of what to present and how. Therefore, a born pedagogue such as Feller does achieve a degree of self-reflection and polish in the first volume that the second volume doesn’t quite match.
Another observation would be that the more difficult material of the second volume was still relatively new, hence less worked-over, at the time of writing. For instance, this reviewer, having already learned real analysis and measure theory from later textbooks such as Royden’s (our review here) or Folland’s (our review here), could not discern anything noteworthy in chapters four and five of the present work. Thus, one would be better off skipping them and focusing on what is unique to probability theory, as for instance, the concepts of infinitely divisible and stable distributions, treated exhaustively here. The same remark would probably apply to Feller’s presentation of the theory of semi-groups of operators in chapters thirteen and fourteen and of harmonic analysis in chapter nineteen.
These topics, we want to suggest, would be better studied in more modern textbooks, to perfect one’s understanding. Thus, what one should concentrate on here would be the applications special to probability theory itself, such as the use of Laplace transforms in chapter fourteen or of characteristic functions in chapter fifteen. If one warns oneself not to be misled by the table of contents to think that everything one needs to know can be found in this second volume, but is judicious about what to pick up here and what to learn elsewhere, it can be a profitable exercise to go through Vol. II. For, as we have indicated, its level is uneven. Section five in chapter one on ‘The persistence of bad luck’ is matchless! If only Feller had not been prevented by his untimely death from bringing the rest of it up to this standard! Three stars.
February 9, 2019
I felt this one not so interesting. I am math hater, was looking for something interesting and easy to understand. I however, found out that, Willian fellers books are good from the perspective of theory and concepts. But ultimate maths is not that easy. Other books(such as Charles M. Grinstead, J. Laurie Snell - Introduction to Probability Second Revised Edition)are even more confusing.
December 11, 2021
Good this book as a recommendation by undergrad supervisor Oscar Sotolongo. I have come back to it in several occasions. Specially to the topic of stable distributions, the equivalent of the normal distribution in the fat tails world.
Displaying 1 - 4 of 4 reviews