Probability: A Graduate Course (Springer Texts in Statistics) by Allan Gut (PDF)

Description

This textbook on the theory of probability starts from the premise that rather than being a purely mathematical discipline, probability theory is an intimate companion of statistics. The book starts with the basic tools, and goes on to cover a number of subjects in detail, including chapters on inequalities, characteristic functions and convergence. This is followed by explanations of the three main subjects in probability: the law of large numbers, the central limit theorem, and the law of the iterated logarithm. After a discussion of generalizations and extensions, the book concludes with an extensive chapter on martingales.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Many of the fine lines in standard mathematics symbols were just not properly scanned into this edition. infinities become weird looking “x” marks, indices of sums/unions often have bounds but not the index variable itself. At least 1 of the first 3 problems in the book has an entire sentence missing (either due to an older edition being scanned or the scan being done poorly) which makes it entirely false. 1 and a half weeks into the semester and I have already wasted so much time trying to figure out what was lost compared to a proper print version. The only redeeming quality is that I (so far) have been able to mostly figure out these types of issues.

⭐I have been using this book for several years, including in an Advanced Probability course as a supplement to Billingsley’s Probability and Measure. On the whole, I found Prof Gut’s book very useful. It is comprehensive and up to date, well organized, and a good source of worked-out examples and hints for proofs and homeworks. For the most part, its proofs are logical, efficient, and clear. As I was thinking of buying the Kindle version of the book as well, I checked the reviews on this site and was shocked to find a review by Diff D that called it a “careless and shoddy book”.After reading that review, I conclude that it is highly unfair. The reviewer Diff D has built up the existence of typos and some trivial mistakes into high crimes. I was particularly struck by Diff D’s claim, in paragraph 4, that the book’s statement that “a sequence of reals converges if and only if every subsequence contains a convergent subsequence” is “laughable”, “appalling”, and “nonsense”. In fact, the statement in question may be somewhat elliptic, but it is basically and obviously correct. Written out in full, it says: “a sequence of reals converges to a real number X if and only if every subsequence contains a subsequence that converges to X”. This “subsequence principle”, unknown to Diff D, is so elementary and easy to prove that Aliprantis and Burkinshaw leave it as an exercise in their Principles book (exercise 8, Ch. 1, Section 3, 2nd Ed; exercise 2, Ch, 1, Section 4, 3rd Ed.). Billingsley presents another version of the same principle in paragraph A10 of the Appendix, Probability and Measure, 3rd Ed. Perhaps Prof. Gut erred in assuming that an intelligent reader would understand that all the convergent subsequences would have to converge to the same limit, otherwise the sequence diverges. In any case, a few lines down on the same paragraph, the author states the principle fully and correctly for random variables, and uses the principle correctly in the subsequent proofs, notwithstanding Diff D’s mocking comments.I advise readers to disregard the inaccurate and unfair review by Diff D and I highly recommend this book.

⭐This is an excellent book. I have found it extremely useful. A must for graduate students in Economics, Finance, Psychology, etc.

⭐I used this book a couple of years ago as a reference for my graduate probability course. I leave this review in response to the only one left so far. Dave D’s remarks not only are false but the lengthy negative review suggest the comments were done out of spite or as some form of self gratitude. This book is intended to soundly cover the proofs of the theorems required for the subject, a task the author believes others have not accomplished (and I must agree). Unfortunately, there is really no such thing as a ‘complete’ proof. You always have to assume that the reader will know certain requirements implied in certain stages of the proof. Still, the author avoids as much as possible statements such as ‘clearly..’ often used by others to shortened proofs (even when results are not yet clear). I think professor Gut did a wonderful job in extending the rather vague proofs frequently found in probability theory textbooks. I suggest readers browse over some proofs from the book and make their own decision.

⭐My review is about the 2005 edition.I wonder often why one can hardly find a bad review in math community?It seems every review done for this book has been excellent and yet after reading a good deal of it I cannot find particularly good words for the book.I would definitely suggest to avoid using this book as a textbook because it suffers from few serious flaws:1. Clarity of exposition. Many dubious statements throughout the book with references to topics lacking precision. Sometimes truly puzzling – for example at the end of the proof of Theorem 5.5.8 on page 224 it stands literaly “do not forget Theorem 2.2.1(iii).” Since there is no such theorem in the textbook and we really do not know what is this applied to, the reader is left scratching his head. It appears there is a use of monotone convergence in the proof that was not mentioned so I am guessing that would be the ghost Theorem 2.2.1 (iii). There is more – at one place the author writes “review the last two sections … and translate the results into the language of mathematics.”This continues through proofs. Take a good look at the utter mess author made out of rather simple proof of Lemma A.4.2. on page 561. One would think this is a work of a confused student copying somebody else’s homework. The author lacks elementary ability to present simple arguments.Another example – Borel’s Normal Number Theorem (theorem 6.9.1 on page 305). At no point the author defines what normal number is (supposedly vague notion of relative frequency should lead us to that), while the proof uses variables Xi which are unidentified, at one point it looks like these are the value of digits while in the proof it is obvious these are Bernoulli variables, of course for those who already know the theorem. Professor Gut just assumes the reader knows what he is talking about “out of his head” without any clearly defined notion.2. “Details left to reader” when it is obvious that author has not proven the statements himself given the missing terms in statements for example. Look at the example in theorem 5.2. of the chapter 5 on page 218. First the author “forgets” to state integrability for r-th moments of variables in the (b) and (c) part of the statement then he ends up in (a)->(b) with one term conveniently vanishing without any explanation (check classic textbook “Probability and Measure” by Billingsley, theorem 16.14 for nontrivial deduction), then in (c)->(a) he uses a proposition 2.6.3 on Stilstjes continuity that indicates he forgot to include the integrability conditions. On the page 221 the author repeats this theorem, this time with convergence in probability instead of almost sure convergence. Of course, the proof is left to reader with a note “as in the previous proof.” He will repeat that yet again in Theorem 5.5.9 with where a part of proof notes “as in the proof of previous two theorems.” :-)Patrick Billingsley in his vastly superior textbook has not forgotten integrability conditions even though he delegated the whole discussion to an optional section on uniform integrability and treated only r = 1 case.Reading mess that is professor Gut’s book makes me appreciate authors like Billingsley.3. Careless deductions when the proofs are available. Just take a look at the subsection “12. Tail Probabilities and Moments” of the Chapter 2 and try to wrestle with missing and careless proofs. Or Lemma 8.1 (Khintchine Inequality) on page 147. Some of the nontrivial details missing author is aware of (“please check this fact”) but others he apparently isn’t – he purports to prove the existence of a constant A_p independent of the coefficients of the function f_n yet he is apparently not aware he did not complete the proof of that independence in the end leaving the reader with nontrivial task. This kind of carelessness in proofs of major theorems is a sure sign of poor quality of this textbook.4. Erroneous statements and claims. For one laughable (or appaling if you like and this is by no means the only one) example take a look at the section 7 of chapter 5 on page 229 that starts with the following nonsense: “It is well known that a sequence of reals converges if and only if every subsequence contains a convergent subsequence.” Weierstrass, this one you missed! This is “well known” only to professor Gut I guess. Here is an easy counterexample: 0,1,0,1,0,1,0,1 … This sequence has the property that every subsequence contains a convergent subsequence but the sequence overall is obviously not convergent. On page 230 after proposition 7.1 the author uses this fallacy of “well known fact” to make sweeping conclusions about convergence in probability, mean convergence, and distribution convergence. Yet again he uses this nonsense on page 237 for more theorems 5.8.5 and 5.8.6! And more later, the whole parts of this textbook will have to be rewritten since the author keeps on referring to this fallacy as “subsequence principle” further on as well for example theorem 5.9.1 (Continuity Theorem on Characteristic Functions) on page 238 has a dubious “sufficiency proof” (accompanied by a gibberish of a “proof” of “lemma 9.1”) that has reference to a flawed theorem 5.8.5. And yet again in the proof Theorem 6.8.3 on page 303, which can be salvaged otherwise. I sincerely hope professor Gut has not published any of this embarrasing stuff in any paper elsewhere. After reading this book I would be seriously concerned about any original claim he makes without exhibiting a proof. I am truly amazed that no editor or reviewer has noticed this.Of course there is something of a “subsequence principle” for convergence in probability as it should be well known to all graduate students of Real Analysis but it has nothing to do with nonsense/ignorance displayed by professor Gut.5. Bogus references. Take a look at the page 281, theorem 4.2 there. One needs to be worry of these kinds of theorems since it is from some obscure paper of the author. To be able to understand the proof author directs a reader to appendix where slow and regularly varying functions are discussed as if there will be some resuls to use. In the first lemma there (Appendix 7, page 566) the first two parts are without proof and reader is directed to “literature,” of course we do not know where exactly except that author listed 4 references. In the third lemma there (page 568) author directs a reader to another lemma (with a proof left to reader to “finish” that would not be hard if not coupled with erroneous statement in the second part of lemma) and to the previously unproven lemma. In fact the second part of lemma 3 is not proven nor even mentioned (except as a fact).I know that a great deal of lousy mathematics books are written only for authors themselves (and sometimes only for the sake of vanity or job promotions) but this one is indeed a comedy. Authors often includes his obscurred results which apparently have not passed proper editing. It looks like his own research jammed in anyway possible.Few other examples of utter carelessness:(a) On page 129 the author “proves” the reverse of stated claim (!), the well known Lyapunov’s inequality albeit with some effort. This would be truly comical if not pitiful since it is directed to new probability students. All in all, logic isn’t the strongest point of this book that happens to wrestle laboriously with contrapositive statements throughout and whenever there is a proof involving a statement with more than one quantor one can rest asure that will be a mess. In fact the author should take even a basic course in propositional logic. Then he would not write something like A <-> (B<->C) while he actually meant (A<->B) & (A<->C). This is precisely what he wrote in the second claim of Lemma 4.1 in the appendix on page 560. One does not write math books like James Joyce unless it is intended to utterly confuse readers. If you want to see how author’s flawed logic makes a complete gibberish of a proof of a well known theorem just read the proof of Kolmogorov’s Three Series Theorem on page 289. It looks like it is written by a graduate student on qualifying examination under pressure. Just appaling that anybody would consider this a textbook presentation. Or another example of author not able to handle the most basic facts from predicate calculus – on page 297 in the “proof of (c)” of theorem 6.6.1 he concludes that negation of statement “for every c statement A holds” is “for every c statement A does not hold.” An error appropriate for undergraduate student or worse. Fortunately the proof does not need “for every” conclusion anyway. Truly astonishing for a graduate textbook to exhibit such a flawed command of the most basic logic.(b) On page 140 the author gives 2-line definition of alpha-quantile where he makes no less than 2 errors – the definition is in reverse and incomplete. The amazing thing is that immediately after the definition he states the theorem (straight out of Valentin Petrov’s well known treatise) about this notion of alpha-quantile and expects the reader to prove it! Apparently the whole section was just carelessly copied without regard for any rigor. This kind of recklessness is rather appalling on the graduate level (a “graduate” course). Does the author expect that a graduate student would simply brush over the statements without critical reading?(c) On page 146 Author introduces well known topic of Rademacher functions. Yet another comedy however benign. Take a good look at his definition and the two figures that follow. First the definition is inconsistent itself since r(t) is defined on [0,1] but then it takes values bigger than 1 in r2(t), …. The figures themselves do not correspond to definition either since there t can be anything between -1 and 1. Should we guess what he intended?(d) On page 563-564 the author “proves” rather elementary Clarkson’s inequality which goes by standard calculus argument until the very end where author makes multiple calculus errors! A good calculus student would notice that the analysis of the first derivative was enough.(e) On page 308 in the proof of theorem 6.9.4 author mentions inversion equivalence formula for counting process of records, unfortunately with a notorious and known flaw – the inequalities are reversed from what they should be. Fortunately he had no need for it in the proof itself…. and so on and on and on …The sad part of this carelessness is that the book’s content is valuable, there are many results rarely found elsewhere thus it would be a valuable read if it were not so reckless. The author himself is undoubtedly well informed specialist in limit theorems (although not for reals :-)).I can grant the author the excuse of the first edition and hopefully this book will be edited properly if it survives. Author should also be very careful about the erroneous statements he makes in theorems and propositions. The rule of thumb should be to prove everything yourself precisely before stating it formally since sometimes forgotten conditions, terms or special cases make it extremely hard to decipher while reading it the first time.This flies in the face of bizzare preface where the author complains about the arrogance and carelessness of others in math community! That is precisely what this book is about. For a fine example of arrogance compare the proofs of Helley’s Selection Theorem in this book and in Kai Lai Chung’s book “A Course in Probability.” Professor Chung spends half a page on page 89 to properly complete the proof after construction of the limit. Professor Gut couldn’t be bothered to do that. Perhaps professor Gut thinks we should already know the famous theorem so why bother. If so perhaps he should have skipped all the first 4 chapters and most of other well known results since the reader is anyway left to references for proper proofs anyway.I am also amazed with the poor editing done with this text in Springer.Responsible: George Casella, Stephen Fienberg, and Ingram Olkin, apparently all “very important” people whose appearance as editors is just that – appearance.Springer’s high priests do not read as long as the writing is looking complicated enough or comes from “reputable” source. It seems this is what happens when the money and vanity is the primary goal of publishing math texts. Just casual reading could reveal many misprints. Take a look at the definition of characteristic function on page 158, it contains three misprints in a single line, two of them in a single expression! Did anybody even try to proofread definitions? Misprints are ALL OVER the place in the notation which adds to the confusion created by the author with his careless presentations. Not all of them are benign (easy to spot) for example on page 195 there is an “extra” logarithm in lognormal density. Another example – on the bottom of page 236 in the second term of the inequality there the author left an “extra” indicator function while in the statement (ii) of the corollary that follows (without a proof of course) he forgot an absolute value sign. Or Anscombe Condition presented on page 346 with mistaken quantors leading to the inequality. Missing terms, “extra” terms, or misplaced quantors can create a particular grief during the reading unless you already know the topics. It destroys the interest of reader, for example I had a keen interest in reading author’s exposition of the so called “Moment Problem” until I actually started reading it – mostly descriptive references to work of others with very little deductions, just barely confined on pages 195-196 with, of course, numerous misprints again. The tract ends up with another dubious theorem (of little interest) copied out from one of the author’s papers with incomplete proof of course, exactly as in the paper where nobody noticed it either. The carelessness continue over well known theorems and results as well. For example the proof of Skorohod’s Representation Theorem on page 258-259 contains no less than 11 misprints (a treasure hunt to find them is needed) which renders it unreadable, nevermind it is already a complete mess as it is, in a similar way many proofs in this book are. The author is simply incapable of consistent and logical presentation of anything, it seems. This includes the problems sets (exercises) and appendices where he keeps on with misprints, erroneous claims, ambiguous statements, missing conditions, jumbled and faulty definitions etc. Just take a look at Appendix 9 (“Functions and Dense Sets”) and try to decipher professor Gut’s line of thought. This is indeed not mathematics but utter confusion and hodgepodge.And at the end, my own lemon award to all the reviewers that have given “excellent” reviews to this book without actually reading it. This is definitely the worst “graduate” book I have seen in a while. It should be rather embarrassing to the author and it is certainly appaling that this is something Springer would publish as a graduate text as if nobody there expected anybody would actually read it. Fortunately lately there are many other choices for a decent graduate probability text so the readers should steer clear away from this one. I pity professor Gut’s students since I am sure that this shoddy and pitiful book is forced upon them.

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