## William Feller's *Probability Theory* - Reviews 5

Below is reviews of the Second Edition of Volume II. For Reviews of the First Edition of Volume I, see THIS LINK. For Reviews of the Second Edition of Volume I, see THIS LINK. For Reviews of the Third Edition of Volume I, see THIS LINK. For Reviews of the First Edition of Volume II, see THIS LINK. |

**5. An introduction to probability theory and its applications Volume 2 (2nd edition) (1971), by William Feller.**

**5.1. Review by: David Kendall.**

*The Mathematical Gazette*

**56**(395) (1972), 65-66.

The second edition of a book does not normally call for a review, but this is clearly an unusual case. Feller's 'second volume' is so exceptionally original and outstandingly important that all who already possess it will wish to know in what respects the new edition differs from the first, and this question your reviewer will attempt to answer; it is not quite enough to say that 626 pages have become 669. ... The book retains its profusion of footnotes, crammed with historical allusions and Fellerian adjectives (which have, however, been slightly varied from those in the first edition).

**5.2. Review by: Kai Lai Chung.**

*American Scientist* **60** (5) (1972), 641.

Volume 1 of this book has long been established as a classic in mathematical treatises and has gone through three different editions since its appearance in 1950. It deals only with a countable sample space in order to waste no time on generalities and to proceed at once to interesting topics. This restriction made it impossible to introduce many basic notions; in particular, any serious discussion of stochastic processes in continuous time is out of the question. Feller made important contributions in the fifties to the latter domain, in what he called "generalized diffusions," now known as "Feller processes." There was general expectation that he would ex pound this in Volume 2, as a fitting sequel to his very successful popularization of Markov chains in Volume 1. It must have come as a surprise to many that he did not choose to do so. Although there are copious allusions to the subject, he limited himself in substance to informal discussions and introductory material (see Chapter 10). When asked about this, he responded by invoking the promise of Volume 3, 4. ... Alas, time must have a stop. He lived to finish this substantial revision of the first edition (1966), and others did the proofreading after his death early in 1970. The rapid completion of the second edition is one more testimony to his dedication to this monumental work.

**5.3. Review by: Mark A Pinsky.**

*SIAM Review* **14** (4) (1972), 662-663.

This is a second edition of the author's monumental Volume II. The manuscript was completed at the time of the author's death in January 1970; the mathematics was checked posthumously by five dedicated colleagues. ... Although several theorems have been corrected and simplified, the distinctive feature of this edition is the improved overall clarity of presentation. The increased abundance of examples, displayed formulas, and diagrams indicates a strong desire to entice an even larger group of mathematics students into the probabilistic way of thinking. ... Feller has maximised the ratio of ideas per page while minimising the ratio of definitions to theorems. Let us hope that this can now be appreciated by a greater number of first readers.

**5.4. Review by: J F C Kingman.**

*Journal of the Royal Statistical Society Series A (General)* **135** (3) (1972), 430.

In the death of William Feller the statistical world has lost one of its most distinguished and delightful mathematicians, and as a parting gift we have a second edition of Volume II of his great 'Introduction'. The first edition was reviewed in this *Journal* in 1967, and none of the ingredients which made it so welcome then is in any way diluted in this revised edition; the range and depth of mathematical insight, the freshness and vigour of exposition, the simplifications of complicated proofs, the illuminating examples, the historical footnotes. ... Professor Feller now belongs to history, but this book will continue to exert a living influence on probabilists for many years to come.

**5.5. Review by: S M Samuels.**

*Technometrics* **15** (2) (1973), 420-421.

Feller completed this second edition in 1970 just before he died. The first edition had appeared in 1966. The organization of the book remains unchanged despite the fact that each one of the chapters has undergone some degree of revision. All of the chapter titles are the same, and, although there are alterations of the lists of sections comprising twelve of the nineteen chapters, every section, even the "new" ones, consists largely of material which was already in the first edition. The second edition is slightly larger than the first: 669 pages versus 626. Most of the extra space consists of clarification of the exposition and of repetitions of material to make the individual chapters even more self-contained - hence directly accessible to the reader - than they were in the first edition. A primary goal in writing the first edition was, for Feller, "to consolidate and unify the general methodology" of probability theory by demonstrating how newer and simpler methods of proof revealed, among various parts of the subject, close connections which were formerly "obscured by complicated methods". (From the introduction to the first edition.) Feller's arguments are not always easy to follow, but they are almost always admirable for their clarity, conciseness and comprehensiveness. The main thrust of the revisions in the second edition is to strengthen even further this quality of the exposition. ... this book is for the benefit of everyone who is "interested primarily in results and facts", rather than the purely theoretical aspects of the subjects. For such reader this book is indispensable, if only because it has in it so many more results and facts than any other.

**5.6. Review by: Kai Lai Chung.**

*The Annals of Probability* **1** (1) (1973), 193-196.

Volume 1 of this work has long been established as a classic in mathematical treatises and has gone through three quite different editions. It is elementary in the sense that it begins at the beginning and presupposes little background. [When it first came out it was much used as a textbook for a first course in probability; however, it is becoming increasingly difficult to do so as the audience has widened but its mathematical preparedness has not.] In fact, it deals only with a countable sample space so that all random variables and distributions are necessarily of the discrete type. This of course does not prevent Feller from going off at various deep ends and discussing some of the most up-to-date topics of interest. The restriction was deliberately imposed in order to waste no time on dull generalities (often indiscriminately referred to as "measure theory") and to proceed at once to significant results. It is clear that many basic concepts cannot even be defined in the restricted context, and any serious exposition of stochastic processes in continuous time would be out of the question. But the latter is precisely where the action is at these days. Feller himself made fundamental contributions in the nineteen-fifties to this area called by him "generalized diffusions" and by others "Feller processes" (precursor of Doob-Hunt-Meyer processes); albeit largely in an analytic form, without "paths," or rather with these only lurking in the background. It was therefore the general expectation that he would take this up in Volume 2 as a main theme. This would be a fitting sequel to his extremely successful popularization of Markov chain theory in Volume 1, acknowledged by himself in the Preface to the first edition. It must have come as a surprise to many that this is not what he did in Volume 2. Although there are recurrent and persistent allusions to bigger and better things to come (witness on page 333 "we are not at this juncture interested in developing a systematic theory"), general discussion of such processes is mostly relegated to heuristic descriptions, preparatory material and footnotes-not too copious. When asked why he chose not to expound his own theory of diffusion, Feller had responded by invoking the future promise of Volume 3, 4, ... . Alas, time has stopped too soon for him and his readers. What he did here, in his characteristic vigour and verve, may be described in his own words as "a systematic exploitation and development of the best available techniques" for analytic problems arising from probability and its applications. ... As is familiar to readers of Volume 1, Feller's style is vivid and personally involved. His discontent with the unlearned, unkempt or unregenerate in the field is evident. He has made enormous efforts to present his favourite topics in the best light as he saw it.

JOC/EFR August 2016

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