What do you think?
Rate this book
A Basic Course in Probability Theory
Introductory Probability is a pleasure to read and provides a fine answer to the How do you construct Brownian motion from scratch, given that you are a competent analyst?
There are at least two ways to develop probability theory. The more familiar path is to treat it as its own discipline, and work from intuitive examples such as coin flips and conundrums such as the Monty Hall problem. An alternative is to first develop measure theory and analysis, and then add interpretation. Bhattacharya and Waymire take the second path. To illustrate the authors' frame of reference, consider the two definitions they give of conditional expectation. The first is as a projection of L2 spaces. The authors rely on the reader to be familiar with Hilbert space operators and at a glance, the connection to probability may not be not apparent. Subsequently, there is a discusssion of Bayes's rule and other relevant probabilistic concepts that lead to a definition of conditional expectation as an adjustment of random outcomes from a finer to a coarser information set.
There are at least two ways to develop probability theory. The more familiar path is to treat it as its own discipline, and work from intuitive examples such as coin flips and conundrums such as the Monty Hall problem. An alternative is to first develop measure theory and analysis, and then add interpretation. Bhattacharya and Waymire take the second path. To illustrate the authors' frame of reference, consider the two definitions they give of conditional expectation. The first is as a projection of L2 spaces. The authors rely on the reader to be familiar with Hilbert space operators and at a glance, the connection to probability may not be not apparent. Subsequently, there is a discusssion of Bayes's rule and other relevant probabilistic concepts that lead to a definition of conditional expectation as an adjustment of random outcomes from a finer to a coarser information set.
Kindle Edition
First published July 27, 2007
5 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 of 1 review
April 7, 2025
The ambition of every serious student in the hard sciences must, of right, be to combine conceptual comprehension of principles with technical facility in carrying out derivations or computations. This general maxim will be even truer in the case of probability theory than for most other disciplines as its philosophical foundations can be obscure while the state of the art has become by now quite advanced from the technical point of view (in many areas of pure mathematics, by contrast, one can get away with paying almost no attention to philosophical matters as it suffices just to grasp a handful of axioms before launching into the hard-core details). This reviewer has previously pointed out some directions in which the beginner may proceed on both fronts: the theoretical, with John Maynard Keynes (our review here), Richard von Mises (our review here) and A.N. Kolmogorov (our reviews here and here), while for the applied and practical, with William Feller’s standby textbook in two volumes (our reviews here and here). But, supposing one has gotten this far already, where to turn next?
The present review is dedicated to Rabi Bhattacharya and Edward C. Waymire’s A Basic Course in Probability Theory (Springer-Verlag, second edition, 2016) – which, notwithstanding its title, proves to be well suited for the graduate student at a more advanced level, who will have completed a beginning-level course in real analysis and is already familiar with elementary ideas of probability (such as may be gained from an upper undergraduate-level course for which Feller’s introductory textbook would be ideal). Our motivation is now to reinforce the concepts of probability theory in greater technical depth and, at the same time, to gain experience with real analysis in the context of some significant applications. Why choose Bhattacharya and Waymire as one’s guides in this endeavor? In their standard expositions, Feller and Doob’s reliance on analytical technique beyond the usual differential and integral calculus is almost non-existent while A.N. Shiryaev’s otherwise excellent graduate-level textbook on probability (our review here) can be a little light in its employment of real analysis.
Bhattacharya-Waymire, on the other hand, will prove ideal for those possessing some mathematical maturity who want to learn about probability, in that it assumes from the start a good acquaintance with real analysis. The level of sophistication in this work is distinctly and unapologetically higher than that of Feller and Doob and comparable to or greater than Shiryaev’s: for instance, Shiryaev has nothing on the Wiener process (in continuous time). The Borel-Cantelli lemma will be an old friend by the time one gets through the present presentation! Be aware, however: while it may be very strong on all technical aspects it is fairly weak on conceptual explanation and delves into no philosophy whatsoever. The authors expect that one will have mastered these things before attempting their work itself.
The first chapter contains a lightning review of facts from real analysis as adapted, as the case may be, to the instance of a probability measure: a very dry and spare exposition but readable if one is already familiar with real analysis. Recommend doing the majority of the problems, which are fairly easy, to familiarize oneself with the significance of the boundedness of measure in a probability space. For instance, a problem on the convergence of the binomial distribution to a Poisson distribution in the limit as the parameter p tends to zero: note how much more satisfying than the usual physicist’s proof the method suggested here is (based on a simple application of the Lebesgue dominated convergence theorem)! One can save time by skipping a few problems that are nothing but elementary results in real analysis (one should have solved them before as homework exercises in Royden or Folland).
Chapter two gets into some basic concepts of probability theory proper: independence, conditional expectation, Markov property. The pace is brisk, but one welcomes here what will emerge as a signature of the work as a whole, the illustration of the ideas through non-trivial examples. The discussion of percolation and the attendant problem are excellent cases in point. The numerous exercises to this chapter are quite instructive.
Chapter three turns to a couple other elementary concepts that are ubiquitous in modern probability theory, those of martingales and stopping times. It is somewhat surprising that, with the aid of Kolmogorov’s and Doob’s maximal inequalities, one can use the idea of stopping times to work out some unexpected and non-trivial statements, such as an explicit formula for the probability in an asymmetric random walk starting from a given point of reaching another point or, with the symmetric random walk, state the property of recurrence (that, no matter where one starts, it eventually reaches every point with probability one). A few other very interesting examples of the techniques introduced in this chapter include the binary multiplicative cascade and the probability of ruin in insurance risk (the Cramér-Lundberg and Sparre Andersen models).
The relatively brief chapters four and five enter into a few classical topics, viz., the central limit theorem, zero-one law, law of large numbers and large deviations. These, along with some nice refinements of the theory such as Berry-Esseen’s estimate of the rate of convergence to a normal distribution or Hoeffding’s inequality, receive efficient proofs through application of the techniques in real analysis the authors have built up.
Many readers, however, will be more familiar with the standard proof of the central limit theorem via characteristic functions, to which the authors turn in chapter six. The necessary preliminaries on Fourier series and Fourier transform can be somewhat repetitive, but there is nothing wrong with the compressed presentation here, which also has the nice feature of attending to further developments such as Bochner’s theorem (a necessary and sufficient condition for a function to be the Fourier transform of a positive measure) and the Chung-Fuchs lemma (a necessary and sufficient condition for recurrence in random walks).
In later chapters, Bhattacharya and Waymire take the text into the direction of rather more advanced ideas, having to do with weak convergence of probability measures in metric spaces. Despite the difficulty one may fear at first, they manage to render the path pleasing by appeal to techniques from real analysis one will have encountered before in an abstract setting but which here acquire a very concrete application. For instance, the embedding of the Hilbert cube. Prohorov’s theorem on tight collections of probability measures reveals itself to be a fairly straightforward exercise involving compactness. A particularly praiseworthy feature is the authors’ treatment of Brownian motion via Wiener measure. Here, their accustomed clarity of exposition really shines, for otherwise the material would be all but impenetrable (from what this reviewer has seen before, elsewhere). Needless to say, the material gets to be rather technical, compared to previous chapters (see, for instance, Kolmogorov’s extension theorem by means of which to define a probability measure in an infinite-dimensional space from its restriction to finite-dimensional cylinders). All the same, it is rewarding. For one realizes that the Wiener process figures not merely as some fancy and arcane formalization of Brownian motion, but is singled out for a distinguished role as being, in effect, a version of the central limit theorem in infinite dimensions. In chapter ten, one will find a derivation of some spectacular results on fine-scale properties of Brownian motion, such as the law of the iterated logarithm. These ideas are extrapolated in chapter eleven to a more general setting by recourse to the strong Markov property and Skorohod’s embedding theorem. After a historical digression in chapter twelve on early work on Brownian motion, the final chapter thirteen takes up the generalization from special examples such as the random walk or Brownian motion to general Markov processes and shows how to prove their convergence to equilibrium. A good worked example would be a birth-death Markov chain with reflection. The exercises pursue a few further cases (asymmetric random walk with reflection, simple random walk with absorption, the Ehrenfest model, Ornstein-Uhlenbeck processes and the dilogarithmic random walk).
About the 196 homework exercises: most are not that hard, none outstandingly difficult. They are meant rather to help one internalize the material presented immediately before in each chapter, either by filling in a gap in a proof or by applying what has just been done to a closely related problem, mutatis mutandis. Let no one contemn these exercises! For no matter how much of a hot shot one may be, when faced with what is to oneself a novel concept, it is necessary to have the humility to descend to the elementary and to trace through its use in a few simple contexts.
Conclusion: an excellent and, indeed, elegant treatment of advanced probability theory, five stars. Brings one up to the frontier of knowledge. Anyone who masters Bhattacharya-Waymire will be in a good position to commence doctoral-level research. There is still a long way to go, let us caution! For modern probability theory is an expansive field and the present work happens to be rather succinct (what constitutes its virtue); thus, it cannot pretend to take one very far into all that one has to know to become a versed practitioner (for instance, the whole theory of 1-parameter semigroups of operators as it pertains to stochastic processes). Fortunately, the two authors have recently issued two installments in the Springer Graduate Texts in Mathematics series, what we presume would be the logical place to follow up.
The present review is dedicated to Rabi Bhattacharya and Edward C. Waymire’s A Basic Course in Probability Theory (Springer-Verlag, second edition, 2016) – which, notwithstanding its title, proves to be well suited for the graduate student at a more advanced level, who will have completed a beginning-level course in real analysis and is already familiar with elementary ideas of probability (such as may be gained from an upper undergraduate-level course for which Feller’s introductory textbook would be ideal). Our motivation is now to reinforce the concepts of probability theory in greater technical depth and, at the same time, to gain experience with real analysis in the context of some significant applications. Why choose Bhattacharya and Waymire as one’s guides in this endeavor? In their standard expositions, Feller and Doob’s reliance on analytical technique beyond the usual differential and integral calculus is almost non-existent while A.N. Shiryaev’s otherwise excellent graduate-level textbook on probability (our review here) can be a little light in its employment of real analysis.
Bhattacharya-Waymire, on the other hand, will prove ideal for those possessing some mathematical maturity who want to learn about probability, in that it assumes from the start a good acquaintance with real analysis. The level of sophistication in this work is distinctly and unapologetically higher than that of Feller and Doob and comparable to or greater than Shiryaev’s: for instance, Shiryaev has nothing on the Wiener process (in continuous time). The Borel-Cantelli lemma will be an old friend by the time one gets through the present presentation! Be aware, however: while it may be very strong on all technical aspects it is fairly weak on conceptual explanation and delves into no philosophy whatsoever. The authors expect that one will have mastered these things before attempting their work itself.
The first chapter contains a lightning review of facts from real analysis as adapted, as the case may be, to the instance of a probability measure: a very dry and spare exposition but readable if one is already familiar with real analysis. Recommend doing the majority of the problems, which are fairly easy, to familiarize oneself with the significance of the boundedness of measure in a probability space. For instance, a problem on the convergence of the binomial distribution to a Poisson distribution in the limit as the parameter p tends to zero: note how much more satisfying than the usual physicist’s proof the method suggested here is (based on a simple application of the Lebesgue dominated convergence theorem)! One can save time by skipping a few problems that are nothing but elementary results in real analysis (one should have solved them before as homework exercises in Royden or Folland).
Chapter two gets into some basic concepts of probability theory proper: independence, conditional expectation, Markov property. The pace is brisk, but one welcomes here what will emerge as a signature of the work as a whole, the illustration of the ideas through non-trivial examples. The discussion of percolation and the attendant problem are excellent cases in point. The numerous exercises to this chapter are quite instructive.
Chapter three turns to a couple other elementary concepts that are ubiquitous in modern probability theory, those of martingales and stopping times. It is somewhat surprising that, with the aid of Kolmogorov’s and Doob’s maximal inequalities, one can use the idea of stopping times to work out some unexpected and non-trivial statements, such as an explicit formula for the probability in an asymmetric random walk starting from a given point of reaching another point or, with the symmetric random walk, state the property of recurrence (that, no matter where one starts, it eventually reaches every point with probability one). A few other very interesting examples of the techniques introduced in this chapter include the binary multiplicative cascade and the probability of ruin in insurance risk (the Cramér-Lundberg and Sparre Andersen models).
The relatively brief chapters four and five enter into a few classical topics, viz., the central limit theorem, zero-one law, law of large numbers and large deviations. These, along with some nice refinements of the theory such as Berry-Esseen’s estimate of the rate of convergence to a normal distribution or Hoeffding’s inequality, receive efficient proofs through application of the techniques in real analysis the authors have built up.
Many readers, however, will be more familiar with the standard proof of the central limit theorem via characteristic functions, to which the authors turn in chapter six. The necessary preliminaries on Fourier series and Fourier transform can be somewhat repetitive, but there is nothing wrong with the compressed presentation here, which also has the nice feature of attending to further developments such as Bochner’s theorem (a necessary and sufficient condition for a function to be the Fourier transform of a positive measure) and the Chung-Fuchs lemma (a necessary and sufficient condition for recurrence in random walks).
In later chapters, Bhattacharya and Waymire take the text into the direction of rather more advanced ideas, having to do with weak convergence of probability measures in metric spaces. Despite the difficulty one may fear at first, they manage to render the path pleasing by appeal to techniques from real analysis one will have encountered before in an abstract setting but which here acquire a very concrete application. For instance, the embedding of the Hilbert cube. Prohorov’s theorem on tight collections of probability measures reveals itself to be a fairly straightforward exercise involving compactness. A particularly praiseworthy feature is the authors’ treatment of Brownian motion via Wiener measure. Here, their accustomed clarity of exposition really shines, for otherwise the material would be all but impenetrable (from what this reviewer has seen before, elsewhere). Needless to say, the material gets to be rather technical, compared to previous chapters (see, for instance, Kolmogorov’s extension theorem by means of which to define a probability measure in an infinite-dimensional space from its restriction to finite-dimensional cylinders). All the same, it is rewarding. For one realizes that the Wiener process figures not merely as some fancy and arcane formalization of Brownian motion, but is singled out for a distinguished role as being, in effect, a version of the central limit theorem in infinite dimensions. In chapter ten, one will find a derivation of some spectacular results on fine-scale properties of Brownian motion, such as the law of the iterated logarithm. These ideas are extrapolated in chapter eleven to a more general setting by recourse to the strong Markov property and Skorohod’s embedding theorem. After a historical digression in chapter twelve on early work on Brownian motion, the final chapter thirteen takes up the generalization from special examples such as the random walk or Brownian motion to general Markov processes and shows how to prove their convergence to equilibrium. A good worked example would be a birth-death Markov chain with reflection. The exercises pursue a few further cases (asymmetric random walk with reflection, simple random walk with absorption, the Ehrenfest model, Ornstein-Uhlenbeck processes and the dilogarithmic random walk).
About the 196 homework exercises: most are not that hard, none outstandingly difficult. They are meant rather to help one internalize the material presented immediately before in each chapter, either by filling in a gap in a proof or by applying what has just been done to a closely related problem, mutatis mutandis. Let no one contemn these exercises! For no matter how much of a hot shot one may be, when faced with what is to oneself a novel concept, it is necessary to have the humility to descend to the elementary and to trace through its use in a few simple contexts.
Conclusion: an excellent and, indeed, elegant treatment of advanced probability theory, five stars. Brings one up to the frontier of knowledge. Anyone who masters Bhattacharya-Waymire will be in a good position to commence doctoral-level research. There is still a long way to go, let us caution! For modern probability theory is an expansive field and the present work happens to be rather succinct (what constitutes its virtue); thus, it cannot pretend to take one very far into all that one has to know to become a versed practitioner (for instance, the whole theory of 1-parameter semigroups of operators as it pertains to stochastic processes). Fortunately, the two authors have recently issued two installments in the Springer Graduate Texts in Mathematics series, what we presume would be the logical place to follow up.
Displaying 1 of 1 review