## William Feller's *Probability Theory* - Reviews 4

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

**4. An introduction to probability theory and its applications Volume 2 (1966), by William Feller.**

**4.1. Review by: U Grenander.**

*Revue de l'Institut International de Statistique / Review of the International Statistical Institute*

**35**(3) (1967), 325-326.

The number of textbooks on probability and statistics is growing at a threatening pace. Many of these books are very similar to each other in their approach and style, but occasionally one encounters exceptions, books that appear to have a lifetime far in excess of the average. In statistics one could think of Harald Cramér's 'Mathematical Methods of Statistics' and Eric Lehmann's 'Testing Statistical Hypotheses'. In probability an example would be William Feller's famous textbook, Volume I. It comes as no surprise that the second volume belongs to the same category. What may be surprising is rather the choice of material and the emphasis that Professor Feller has given to different topics. In comparison with the majority of textbooks on probability, this is fairly sophisticated mathematically. Nevertheless, it is very concrete in style, stressing the actual meaning of the many stochastic models that are described and analyzed. After reading the book it should be obvious to the reader that much of current work in probability theory has a direct motivation from many areas of applications. Indeed, the author has illustrated this through an abundance of special cases, examples and problems for the reader, and the result is most convincing. At the same time he has brought out the historical continuity of the subject by pointing backwards to pioneers whose work would otherwise not always be appreciated at its true value. At the same time he establishes contact with neighbouring areas of mathematics, especially functional analysis. This has resulted in a book that is very much alive, which does not necessarily mean that it is easy to read. On the contrary, there are parts that will tax the endurance and patience of the reader.

**4.2. Review by: R M Loynes.**

*Biometrika* **54** (1/2) (1967), 336-337.

For some years now rumours have been in circulation that Volume II of Professor Feller's book, impatiently awaited by the whole probabalistic world, was just on the point of appearing, and these rumours were accompanied by obscure and rather sinister warnings that it was 'not what you would expect'. Finally it had appeared, and it is probably fair to say that these warning were justified, but the result is much more interesting than if it were not. Probably the first impression one gets on reading even a small part is of freshness and vigour. New or improved proofs and illuminating remarks abound, and although not even Professor Feller can make your reviewer enthusiastic about some of the computations involved in the limit theorems, nevertheless there are many topics, such as for example the exponential distribution, about which one would have thought it impossible to be very original, where either something new is in fact said or at least old material is presented in a striking way.

**4.3. Review by: David Siegmund.**

*SIAM Review* **11** (2) (1969), 295-297.

This book is the second volume of William Feller's 'An Introduction to Probability Theory and its Applications', the first volume of which appeared in 1950 and has been revised twice (1957, 1968). Volume I treated probability in discrete sample spaces, a limitation imposed by the author in order to make his book accessible to an audience of non-specialists. It was an acknowledged success in this regard, and although the second volume is not formally bound by any such restrictions, the author has made a considerable effort to write a book which requires minimal mathematical prerequisites of the reader. I suspect that knowledge of advanced calculus, some familiarity with Volume 1, and perseverance are the essential prerequisites to most of Volume 2. ... Like its predecessor, this volume is permeated with concrete examples and applications of general results. This characteristic distinguishes Professor Feller's book from most mathematics texts and is certainly one of its most valuable features. The exposition is original and elegant. ... This is an excellent book, which serious students of probability will find both stimulating and informative.

**4.4. Review by: Joshua Chover.**

*Journal of the American Statistical Association* **62** (319) (1967), 1070-1071.

The long awaited Volume II of William Feller's 'Introduction to Probability Theory and Its Applications' has finally appeared. I think that, in whole or in part, it is must reading for anyone who has serious interest in the theory or applications of probability. Like Volume I of the same title, it is no ordinary text, but a very rich personal blend of authoritative discussions of fascinating examples and of various levels of theory. The arguments and proofs presented have been polished to a great degree of compactness and elegance, many having a strong intuitive appeal. The virtue of this compression is, often, that the outlines of a topic emerge without the blurring caused by long and detailed proofs. This makes fresh, lively reading for one who has already had some experience with methods of proof, or for one who doesn't need to be convinced rigorously but merely wants to see the main picture. The debit of the compression is that a reader desiring complete rigor must often supply omitted steps, sometimes subtle ones; and in this respect the going can be rough for a novice. The applications described are mostly to real problems and truly illuminate the power of the theory. Moreover the author often relates examples and theoretical results which may have seemed far apart, and shows surprising syntheses.

**4.5. Review by: Frank Spitzer.**

*The Annals of Mathematical Statistics* **38** (3) (1967), 951-952.

This is the long awaited continuation of the author's famous 'volume one'. It begins with three chapters concerning probability densities on one and higher dimensional Euclidean space. They fully recapture the spirit of volume one; indeed this material would well supplement a beginner's course from volume one, which deals only with discrete probability spaces. And just as in volume one the author delights in giving many deceptively simple results which tease the probabilistic intuition or which would require sophisticated proof if viewed outside their natural probabilistic context. The atmosphere of the text turns more austere in Chapters 4 and 5 which introduce measure on Euclidean space in fair detail, and on an abstract probability space in elegant outline (the usual extension theorems, as well as the results of Fubini and Radon-Nikodym are stated without proof). ...

**4.6. Review by: D H Ward.**

*Journal of the Royal Statistical Society Series D (The Statistician) ***17** (2) (1967), 197-198.

The first volume of this book, published in 1950, was, and is, of value to many statisticians. It aroused an interest in probability theory as a branch of mathematics; it gave elegant and comprehensible proofs as well as simpler explanations of the theorems; the examples covered a wide field of practical problems; the book drew attention to many paradoxes and misunderstandings. Volume II was eagerly awaited during the early fifties, since the first volume limited itself to discrete sampling spaces. In the late sixties, how useful is this second volume to statisticians? It claims to be written in the same style as the first except that it involves harder mathematics. But whereas the first volume, in spite of many strolls down side alleys, did seem to have a main theme, the second volume continually repeats itself, and it is difficult to find a way through it, in spite of the author's detailed account at the head of each chapter, of its inter-relations with other chapters. ... To the mathematical statistician, the book may prove a useful source of elegant probability theory, but the topic he wants to study may be difficult to locate, or dealt with in several parts of the book. The applied statistician will have great difficulty in finding the relevant section, and locating the earlier sections on which the vital section depends. Those who found Volume I valuable will probably be disappointed in Volume II - it is not written for the same level of reader. Those who have not read Volume I are strongly advised to study Volume I first. It is as good as Volume II for extending one's grasp of what probability means and what the various limit theorems say and do not say, and contains far more usable applications.

**4.7. Review by: Frank E Grubbs.**

*Technometrics* **9** (2) (1967), 342.

It has been 26 years since Volume I of 'An Introduction to Probability Theory and Its Applications' first appeared in print. Volume I has fulfilled a unique role in its area and has served as a rather complete or "solid" reference book for basic, general probability theory with illustrative examples in the various physical sciences. Volume II, conceived by the author, William Feller, as a companion to Volume I, is clearly a welcome, although more difficult but modern, book on basic probability. Volume II goes very much deeper into the subjects of probability, probability distribution theory of stochastic processes, basic limit theorems, infinitely divisible distributions, Markov processes, renewal theory, random walks, Laplace transforms, characteristic functions, Fourier analysis methods and harmonic analysis. Along with the times, basic measure theory is introduced and employed to provide a more solid foundation for proofs of probability theorems, etc., although as Feller says "... The main function of measure theory ... is to justify formal operations and passages to the limit. ... (although) readers interested primarily in practical results will ... not feel any need for measure theory." Indeed, the chapters are rather self-contained so that many readers in probability could select and gain a more competent view of their own particular interests in probability theory and its applications. Volume II is not for beginners in probability theory and it would help considerably to have studied Volume I carefully before attacking Volume II, although this is not an absolutely necessary background, since Volume II itself is rather self-contained really. Volume II, written in much the same style as Volume I, is for the more advanced students of probability; it is more rigorous, elegant and amounts to a highly technical, compressed compendium, so to speak, although it is lucid nevertheless. The reviewer would say that here we have an excellent text for serious probability students and scientific investigators in the physical sciences (but also other fields), who have need of skilled training in probability and stochastic processes generally.

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

*Journal of the Royal Statistical Society Series A (General)* **130** (1) (1967), 109.

It is now 16 years since Professor Feller published the first volume of his Introduction, which has by now firmly established itself as by far the best mathematical introduction to the theory of probability. But its characteristic clarity and coherence were due at least in part to the restriction to discrete sample spaces, and the more difficult, but no less important, problems connected with continuous sample spaces were deferred to the projected second volume. It is this long-awaited sequel which has now appeared. One's first impression is of the extraordinary amount and variety of the material presented, and of its unorthodox and stimulating arrangement. ... All topics are expounded with clarity and with a wide range of examples. The mathematics is rarely excessive or inelegant, and the book should be accessible to all mathematicians and (with judicious skipping) to less mathematical statisticians. Altogether, this is a book entirely worthy of a place alongside its elder brother. Probabilists have been looking forward to it for a long time; they will not be disappointed.

**4.9. Review by: David A Freedman.**

*Econometrica* **35** (3/4) (1967), 563-564.

Volume I was an enormously successful introduction to probability theory in the discrete case. Volume II does away with this restriction. It is like Volume I in approach and at least as good. What makes Fellers' style so attractive? I think the best short answer is variety. Large parts of Volume II are accessible to advanced undergraduates and interesting to professional mathematicians. The presentation is rigorous but can be studied with profit by someone who neither knows or cares what a Borel set is. Theorems are sometimes proved several times by quite different methods; interesting special cases are often treated independently. Discussions are backed up with examples and problems (answers at the back of the book) in abundance. Impressionistic sampling yielded examples on order statistics, distribution od first significant digits, Bessel functions, gene drifts, and problems on the Holtsmark distribution for stars, the effect of a traffic island, hitting probabilities in random walks, lost calls at a telephone.

**4.10. Review by: A T Bharucha-Reid.**

*The American Mathematical Monthly* **74** (4) (1967), 460.

The first volume of the work under review was published in 1950, and a second edition appeared in 1957. In Volume I, Professor Feller presented a rigorous introduction to discrete probability theory (as well as to certain types of stochastic processes) which was warmly received by a large general readership whose mathematical background was not too extensive, as well as by serious students of probability. Volume II is devoted, in the main, to continuous sample spaces; and the reader is required to have a knowledge and appreciation of the results and techniques of analysis (hard and soft). ... This book is recommended as a text or supplementary text for courses in advanced probability, and as a rich source of material for seminars devoted to topics in probability and stochastic processes. Volume I and II of Professor Feller's book constitute a unique contribution to the literature of probability theory.

**4.11. Review by: D J Bartholomew.**

*Journal of the Royal Statistical Society Series C (Applied Statistics)* **16** (2) (1967), 177-179.

After 16 years of waiting Feller's Volume II has appeared and it amply fulfils the expectations of those who have grown up with Volume I. The only regret that readers of this book are likely to feel is that it has been so long delayed. The high standard of exposition maintained throughout these volumes evidently demands much rewriting and reflection. However, at the risk of appearing irreverent, it is interesting to speculate that something worthwhile would have been gained by publishing a less polished version five or even ten years ago. Posterity may judge otherwise but students of recent generations can be forgiven if they feel thus. ... The fact that Volume II deals with continuous sample spaces means that the mathematics is more difficult than in Volume I. Inevitably this means measure theory which is liable to induce mathematical paralysis in many of those brought up before the days of the "new mathematics". ... The text is written in the same direct and lucid style familiar to readers of Volume I. The combination of informal intuitive argument and mathematical rigour makes the book readable by layman and mathematicians alike. It is not common in mathematical writing to find such overt enthusiasm for the subject as is conveyed by meeting the words "glorious", "famous" and "sensation" in the course of one footnote on p. 421. In the hands of lesser men such language would be irritating but here it captures the spirit of a book which is meant to be enjoyed. ... Perhaps the major problem of the statistical world today is the growing gulf between applied and theoretical (mathematical!) statisticians. This division is tending to harden as journals and societies increasingly cater for specialized interests. At its worst it is found, on the one hand, in that kind of arrogance which assigns a superior status to mathematical abstraction, and on the other, in an inverted snobbery, born of ignorance, which refuses to recognize the power and economy of mathematical argument. Communication across this gulf is hampered by a lack of persons who are able to speak both languages. Professor Feller is one of those who is able to command the respect of both parties even though his work has been predominantly mathematical. It is to be hoped that his writing will not only be admired but emulated so that common ground and co-operation may again be established between the two branches of the subject. Without collaboration "pure theory" and "pure application" become sterile.

**4.12. Review by: D V Lindley.**

*The Mathematical Gazette* **51** (377) (1967), 263-265.

Volume I of this book appeared in 1950, sixteen years ago. It was, and still is, a great success. The reasons for this success, apart from the obvious essential that the book was well-written, were two in number. Firstly, it combined a mathematically rigorous approach to the subject with a genuine interest in the applications. Thus the book could be read with profit and pleasure both by the mathematician and by the scientist who was interested in using probability ideas. Certainly both types of reader would skip passages, but the continuity was not destroyed by this selection. The second reason for its success was that it combined the merits of a good text-book with some of the features of a well-written research paper. The student was able to under- stand whilst the teacher was refreshed by the novelty of the material. (It is not always recognized how easy it is to ruin a course by boredom of the instructor.) The most striking example of this novelty was the approach to Markov chain theory based on considerations of the states of the chain, rather than on formal matrix theory. But Volume I had a serious defect: it dealt only with discrete probability spaces. This meant that many probability topics were omitted because they involved continuity considerations. The reason was obvious: the mathematics of continuous systems involves more sophisticated material than discrete ones, and the whole apparatus of measure theory has to be invoked. The author rightly felt that these ideas should be postponed until the student had a sound appreciation of probability in the simpler, discrete situation. Thus, almost the first question one asked on seeing volume I was "when will Volume II appear?" Now at last it has arrived: the event for which the statistical profession has been waiting for sixteen years has happened. It is very hard to repeat a success. Has Professor Feller succeeded? Your reviewer approaches this judgement with some trepidation. ... Bearing in mind that the material of this volume cannot interest as many people as did that of the first, I predict that the second volume will repeat the success of the first ... These two volumes should be in every mathematics library. They provide a fine testimony to the elegance and power of modern probability theory and its width of applications.

**4.13. Review by: S Orey.**

*Mathematical Reviews* MR0210154 **(35 #1048)**

This is the sequel to the popular first volume [1950; second edition, 1957] but it is designed so that it can be used independently. The reader of this book need not have any prior knowledge of probability beyond a few basic definitions, say as given in the first chapter of the first volume. Indeed it is the author's aim, admirably realized, to be of interest to a diverse audience, ranging from novice to expert. The book has a rich texture, derived from the wealth of problems treated as applications or illustrations of the theory. The striking aspect is the apparent ease and elegance with which these problems are dispatched, frequently making obsolete the original treatment given in the research literature.

JOC/EFR August 2016

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