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Graduate Texts in Mathematics #95
Probability
This book contains a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, ergodic theory, weak convergence of probability measures, stationary stochastic processes, and the Kalman-Bucy filter. Many examples are discussed in detail, and there are a large number of exercises. The book is accessible to advanced undergraduates and can be used as a text for self-study. This new edition contains substantial revisions and updated references. The reader will find a deeper study of topics such as the distance between probability measures, metrization of weak convergence, and contiguity of probability measures. Proofs for a number of some important results which were merely stated in the first edition have been added. The author included new material on the probability of large deviations, and on the central limit theorem for sums of dependent random variables.
- GenresMathematics
624 pages, Hardcover
First published January 1, 1980
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February 6, 2022
The theory of probability, like any mature discipline, has a more theoretical and conceptual side, on the one hand, and a more applied and computational side, on the other. What is of interest, though, is that once one has the right, or at least adequate concepts and definitions at one’s disposal and has fleshed out the basic formalism, the latter can, as it were, take on a life of its own. For the equations are whatever they are, independent of the logical path one traverses to arrive at them. By the late twentieth century, moreover, the technicalities of probability theory had been elaborated into a high art; we have gone far beyond the simple card games that stimulated its early flourishing to a strange world populated by marvelous creatures, such as infinite-dimensional function spaces and the arcane measures defined on them.
For those anxious to get up to speed in the field in its modern dress, the esteemed Russian mathematician A.N. Shiryaev has written a fine, almost ideal exposition originally offered as a course of lectures at the prestigious Steklov institute located at the Moscow State University and now translated into English and published by the Springer Verlag in its series of graduate textbooks in mathematics (we intend to review the second edition released in 1996). Handsomely typeset in the font to which old-timers will be accustomed and which somehow the latest technology doesn’t seem to be able to replicate at a reasonable cost (glance for comparison at the painfully cheap typesetting and inexpert layout in Barbara MacCluer’s Elementary Functional Analysis from the same GTM series, printed in 2009), clean notation, crisp style.
As to content: chap i starts out with elementary notions in probabilistic models with only finitely many possible outcomes, among them the idea of sample points, events, combination of events via logical connectives, random variables, independence and conditional expectation with respect to a decomposition. All these terms are lucidly illustrated with worked examples such as the familiar coin tosses, binomial distribution, multinomial or hypergeometric distribution, Bayes’ theorem and so forth. Chap i culminates in careful derivations of the law of large numbers, Chebyshev’s inequality, random walks, probability of ruin, limit theorems for the Bernoulli scheme (local, de Moivre-Laplace and Poisson), and the ergodic theorem for Markovian chains.
Chap ii gets into some of the more advanced techniques required to handle the case of infinitely many possible outcomes. Here, Shiryaev concisely states Kolmogorov’s axioms and revisits the definition of a random variable, rendered non-trivial by the complications surrounding real analysis in the continuum. For instance, in order to define conditional expectation one appeals to the Radon-Nikodým theorem. The procedure invites reflection on the question, what is a decomposition really? The space of elementary events (normally designated Ω) ought to be viewed as the spectrum of an algebra of observables. Now, decomposition corresponds to forgetting about some of them so clearly the reduced spectrum prescinding from these will be coarser than the one with which one begins. But if we aren’t entitled to know the values of some closely-held observables, we must update all of our probability estimates to reflect this degree of ignorance, and this is precisely what the conditional expectation does.
In chap iii Shiryaev takes up the problem of convergence properties of families of probability measures. The heart of the chapter consists in a detailed proof of the celebrated central limit theorem (via the method of characteristic functions), both in the classical case observing the Lindeberg condition and in the case of non-classical conditions. The remainder of chap ii covers even more advanced topics such as infinitely divisible and stable distributions, metrizability of weak convergence, the relation between weak convergence and almost sure convergence, Kakutani-Hellinger distance, contiguity and entire asymptotic separation of probability measures (nice because it gives the reader occasion to see some non-trivial applications of absolute continuity or singularity of two measures with respect to each other which he will have met most likely without any exemplification in an introductory course on measure theory) and lastly, the rapidity of convergence in the central limit and Poisson’s theorems. One subject this reviewer wishes to toss out for consideration would be Prokhorov’s theorem, which provides a necessary and sufficient condition for relative compactness of a sequence of measures – of interest to physicists in that the reason why Feynman’s path integral remains so recalcitrant is that when working with probability amplitudes one no longer has anything like the criterion of tightness of a family of probability measures by means of which to enforce weak convergence on a subsequence.
Chap iv recurs to sequences and sums of random variables in order to analyze them from another point of view, namely, what may we infer if we are willing to postulate pairwise independence of the random variables? Issues in the surprising zero-or-one law, strong law of large numbers and law of the iterated logarithm.
The techniques so far introduced are certainly powerful enough to yield interesting results for stochastic processes in discrete time, in chap v, stationary (strict sense) random sequences and ergodic theory while in chap vi, stationary (wide sense) random sequences (Fourier analysis, Wold’s expansion) and martingales. Here, Shiryaev teaches us some basics having to do with stopping times, useless of course to the statistical physicist but a core topic in mathematical finance – showing how the latter field challenges mathematicians’ ingenuity to invent methods to address problems of a nature that would not be suggested by a study of the natural world. Shiryaev concludes in chap vii with a classification of states in Markovian chains over a finite probability space. Satisfying, rich enough to be non-trivial though obviously a piece of cake compared to what people investigate nowadays in continuous time in infinite probability spaces etc.
On the homework problems (around 225 of them across the seven chapters) – for the most part, not too difficult although as this reviewer recalls a few do demand more persistence than American (not Russian) students are wont to display.
Five stars amply deserved although this reviewer was tempted at first to award only four. Why? Exhaustive and technical but overly meager in its conceptual justification (no philosophy of what probability is) – a four-page introduction, more of an annotated bibliography, scarcely suffices, and then one launches immediately into a spare definition-theorem-proof format. Quick comparison with Feller and Doob’s old standbys: certainly the familiarity with graduate-level real analysis Shiryaev presupposes flies far above that of either of these two canonical authors (even in Feller’s second volume). But at least Feller’s first volume, if not so much Doob’s, adopts a more leisurely pace suitable for the beginner.
For those anxious to get up to speed in the field in its modern dress, the esteemed Russian mathematician A.N. Shiryaev has written a fine, almost ideal exposition originally offered as a course of lectures at the prestigious Steklov institute located at the Moscow State University and now translated into English and published by the Springer Verlag in its series of graduate textbooks in mathematics (we intend to review the second edition released in 1996). Handsomely typeset in the font to which old-timers will be accustomed and which somehow the latest technology doesn’t seem to be able to replicate at a reasonable cost (glance for comparison at the painfully cheap typesetting and inexpert layout in Barbara MacCluer’s Elementary Functional Analysis from the same GTM series, printed in 2009), clean notation, crisp style.
As to content: chap i starts out with elementary notions in probabilistic models with only finitely many possible outcomes, among them the idea of sample points, events, combination of events via logical connectives, random variables, independence and conditional expectation with respect to a decomposition. All these terms are lucidly illustrated with worked examples such as the familiar coin tosses, binomial distribution, multinomial or hypergeometric distribution, Bayes’ theorem and so forth. Chap i culminates in careful derivations of the law of large numbers, Chebyshev’s inequality, random walks, probability of ruin, limit theorems for the Bernoulli scheme (local, de Moivre-Laplace and Poisson), and the ergodic theorem for Markovian chains.
Chap ii gets into some of the more advanced techniques required to handle the case of infinitely many possible outcomes. Here, Shiryaev concisely states Kolmogorov’s axioms and revisits the definition of a random variable, rendered non-trivial by the complications surrounding real analysis in the continuum. For instance, in order to define conditional expectation one appeals to the Radon-Nikodým theorem. The procedure invites reflection on the question, what is a decomposition really? The space of elementary events (normally designated Ω) ought to be viewed as the spectrum of an algebra of observables. Now, decomposition corresponds to forgetting about some of them so clearly the reduced spectrum prescinding from these will be coarser than the one with which one begins. But if we aren’t entitled to know the values of some closely-held observables, we must update all of our probability estimates to reflect this degree of ignorance, and this is precisely what the conditional expectation does.
In chap iii Shiryaev takes up the problem of convergence properties of families of probability measures. The heart of the chapter consists in a detailed proof of the celebrated central limit theorem (via the method of characteristic functions), both in the classical case observing the Lindeberg condition and in the case of non-classical conditions. The remainder of chap ii covers even more advanced topics such as infinitely divisible and stable distributions, metrizability of weak convergence, the relation between weak convergence and almost sure convergence, Kakutani-Hellinger distance, contiguity and entire asymptotic separation of probability measures (nice because it gives the reader occasion to see some non-trivial applications of absolute continuity or singularity of two measures with respect to each other which he will have met most likely without any exemplification in an introductory course on measure theory) and lastly, the rapidity of convergence in the central limit and Poisson’s theorems. One subject this reviewer wishes to toss out for consideration would be Prokhorov’s theorem, which provides a necessary and sufficient condition for relative compactness of a sequence of measures – of interest to physicists in that the reason why Feynman’s path integral remains so recalcitrant is that when working with probability amplitudes one no longer has anything like the criterion of tightness of a family of probability measures by means of which to enforce weak convergence on a subsequence.
Chap iv recurs to sequences and sums of random variables in order to analyze them from another point of view, namely, what may we infer if we are willing to postulate pairwise independence of the random variables? Issues in the surprising zero-or-one law, strong law of large numbers and law of the iterated logarithm.
The techniques so far introduced are certainly powerful enough to yield interesting results for stochastic processes in discrete time, in chap v, stationary (strict sense) random sequences and ergodic theory while in chap vi, stationary (wide sense) random sequences (Fourier analysis, Wold’s expansion) and martingales. Here, Shiryaev teaches us some basics having to do with stopping times, useless of course to the statistical physicist but a core topic in mathematical finance – showing how the latter field challenges mathematicians’ ingenuity to invent methods to address problems of a nature that would not be suggested by a study of the natural world. Shiryaev concludes in chap vii with a classification of states in Markovian chains over a finite probability space. Satisfying, rich enough to be non-trivial though obviously a piece of cake compared to what people investigate nowadays in continuous time in infinite probability spaces etc.
On the homework problems (around 225 of them across the seven chapters) – for the most part, not too difficult although as this reviewer recalls a few do demand more persistence than American (not Russian) students are wont to display.
Five stars amply deserved although this reviewer was tempted at first to award only four. Why? Exhaustive and technical but overly meager in its conceptual justification (no philosophy of what probability is) – a four-page introduction, more of an annotated bibliography, scarcely suffices, and then one launches immediately into a spare definition-theorem-proof format. Quick comparison with Feller and Doob’s old standbys: certainly the familiarity with graduate-level real analysis Shiryaev presupposes flies far above that of either of these two canonical authors (even in Feller’s second volume). But at least Feller’s first volume, if not so much Doob’s, adopts a more leisurely pace suitable for the beginner.
December 27, 2011
Great textbook. Not the easiest to read at times, but I definitely recommend it.
January 26, 2014
This book is terrific. I used this book to refresh what I learned at graduate school. With two-quarter course on Probability, I still get a lot of new insights from this book.
Displaying 1 - 3 of 3 reviews