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Stochastic Processes
The theory of stochastic processes has developed so much in the last twenty years that the need for a systematic account of the subject has been felt, particularly by students and instructors of probability. This book fills that need. While even elementary definitions and theorems are stated in detail, this is not recommended as a first text in probability and there has been no compromise with the mathematics of probability. Since readers complained that omission of certain mathematical detail increased the obscurity of the subject, the text contains various mathematical points that might otherwise seem extraneous. A supplement includes a treatment of the various aspects of measure theory. A chapter on the specialized problem of prediction theory has also been included and references to the literature and historical remarks have been collected in the Appendix.
- GenresMathematics
664 pages, Paperback
First published January 1, 1953
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June 29, 2025
The decades around World War II were a golden age for probability theory, when all the tools of modern real analysis began to find significant application there. Probably none was more influential in forwarding this development than the American mathematician Joseph L. Doob, who entered the field shortly after the publication of A.N. Kolmogorov’s groundbreaking treatise in 1932 (see our reviews of the English translation, here, resp. the German original, here). For, as a prescient young post-doctoral fellow, Doob perceived the potential inherent in Kolmogorov’s programmatic manifesto which, nonetheless, remains somewhat sketchy. Nowhere would this statement be truer than in the theory of stochastic processes. The theoretical physicist Albert Einstein’s revolutionary 1905 paper on Brownian motion may be said to mark its inception as a mature discipline, but Einstein’s heuristic considerations stand in great need of a secure mathematical foundation. Now, Norbert Wiener discovered a suitable rigorous formulation of Brownian motion in 1918 but much hard work remained to be done in order to understand it better. In the 1930’s, armed with Jean Ville’s concept of a martingale and with notions of topology and convergence drawn from functional analysis, probabilists were released to range freely in a whole new world of their own devising.
The present basic monograph, Stochastic Processes (Wiley, 1953), was conceived and written by Doob himself as the first systematic treatment in textbook form of the newly won theory. A stochastic process is nothing but a collection of random variables ordered by a temporal index, whether discrete or continuous. Accordingly, a stochastic process can describe the evolution of a physical system consisting, generally speaking, in a signal of interest accompanied by unavoidable noise, and so acquires major technological importance. During the period in question, the field of long-distance communications by telephone wire or radio was beginning to flourish under the impact of modern electronics. Engineers of the time were becoming acutely concerned with typical problems in signals processing, such as how best to reconstruct the intended message from the noisy signal received, or how best to predict future events on the basis of what is known thus far (as in queuing theory)—just the sort of thing for which the theory of stochastic processes is well suited to address!
Overview: the table of contents lists about all the standard topics one would expect in a textbook of this nature. The introductory chapter one is too elementary and not really the place to learn the concepts for the first time. In his preface, Doob apologizes for the degree of mathematical rigor to which he resorts, but anyone who has passed a graduate course in real analysis will feel perfectly comfortable. Chapter two scarcely gets beyond a slew of definitions concerning various kinds of stochastic processes that will be investigated in later chapters: processes with mutually independent random variables in chapter three, or with mutually uncorrelated increments or orthogonal random variables in chapter four. The next two chapters are devoted to the important special case of Markov processes with discrete resp. continuous parameter, going methodically through all the permutations (chains with finitely many states versus continuous state spaces, stationary or not etc.). The principal results here are the law of large numbers and the central limit theorem. In chapter seven, convergence theorems for martingales resp. semi-martingales, sums of independent random variables and, as an application, the strong law of large numbers.
More advanced topics begin to appear in chapter eight with the theory of processes with independent increments, as in Brownian motion and Poisson processes, or with orthogonal increments, leading to the subject of stochastic integrals, which provide a formalism in which the idea of obtaining a random variable by integrating over the differential increments of the process in question makes sense. A good example of what one might be aiming for in all this would be a Fourier transform (initially defined formally, but then recast as a rigorous stochastic integral). Fourier analysis turns out to be particularly serviceable in the case of stationary processes, whether of a discrete parameter (chapter ten) or of a continuous parameter (chapter eleven). Here, in either case one seeks a spectral representation in terms of the covariance function. Given the spectral representation, one can perform linear operations with stationary processes as specified by a gain function—corresponding, in special cases, to ordinary differentiation or to integral averaging (that is, also referred to as filtering). The final chapter twelve covers linear least squares prediction of stationary (wide sense) processes, evidently an important application of the theory.
Doob’s exposition may be a little more leisurely and easier to follow than Feller’s (see our review here), but regrettably is virtually free of examples. It could also be faulted for being unilluminating: throughout, Doob’s proofs though highly technical do not appear to be very deep. The reason for this is that he stays at too high a level of generality. The derivations are clean and efficient, to be sure, but amount merely to a very general structural characterization of stochastic processes (akin to, in measure theory, the canonical decomposition of an arbitrary measure into its pure point, absolutely continuous and singular parts). In the absence of good examples, the danger remains that such general statements fail to acquire much concrete meaning in the mind of the student. What one will not find here are any refined results in the analysis of Brownian motion, such as the arcsine law or the law of the iterated logarithm.
Three stars: in the present work, there is altogether too much merely formal development without any exemplification in real-world use cases—an outcome which may be judged perhaps understandable given that, when Doob was writing, the theory had only just completed its initial stage of development (illustrating, thus, an unfortunate tendency in pure mathematics to run ahead with the formal elaboration in default of sufficient exploration of examples). The absence of homework exercises, moreover, hinders any deeper comprehension. This reviewer plans to go on next to another, decidedly recent work on stochastic processes by Rabi N. Bhattacharya and Edward C. Waymire, Stochastic Processes with Applications (SIAM Classics in Mathematics, 2009); he was pleasantly surprised by these authors’ elegant A Basic Course in Probability Theory (Springer-Verlag, second edition, 2016, our review here) and hopes that they have continued to deploy their crisp and clear modern style of exposition in their much longer comprehensive work on stochastic processes, which appears to be pitched to a reader in command, indeed, of a certain degree of mathematical maturity but seeking a good intuitive feel for the outlines of the subject more than unrelenting and painstaking rigor. So, be on the watch for a forthcoming review!
The present basic monograph, Stochastic Processes (Wiley, 1953), was conceived and written by Doob himself as the first systematic treatment in textbook form of the newly won theory. A stochastic process is nothing but a collection of random variables ordered by a temporal index, whether discrete or continuous. Accordingly, a stochastic process can describe the evolution of a physical system consisting, generally speaking, in a signal of interest accompanied by unavoidable noise, and so acquires major technological importance. During the period in question, the field of long-distance communications by telephone wire or radio was beginning to flourish under the impact of modern electronics. Engineers of the time were becoming acutely concerned with typical problems in signals processing, such as how best to reconstruct the intended message from the noisy signal received, or how best to predict future events on the basis of what is known thus far (as in queuing theory)—just the sort of thing for which the theory of stochastic processes is well suited to address!
Overview: the table of contents lists about all the standard topics one would expect in a textbook of this nature. The introductory chapter one is too elementary and not really the place to learn the concepts for the first time. In his preface, Doob apologizes for the degree of mathematical rigor to which he resorts, but anyone who has passed a graduate course in real analysis will feel perfectly comfortable. Chapter two scarcely gets beyond a slew of definitions concerning various kinds of stochastic processes that will be investigated in later chapters: processes with mutually independent random variables in chapter three, or with mutually uncorrelated increments or orthogonal random variables in chapter four. The next two chapters are devoted to the important special case of Markov processes with discrete resp. continuous parameter, going methodically through all the permutations (chains with finitely many states versus continuous state spaces, stationary or not etc.). The principal results here are the law of large numbers and the central limit theorem. In chapter seven, convergence theorems for martingales resp. semi-martingales, sums of independent random variables and, as an application, the strong law of large numbers.
More advanced topics begin to appear in chapter eight with the theory of processes with independent increments, as in Brownian motion and Poisson processes, or with orthogonal increments, leading to the subject of stochastic integrals, which provide a formalism in which the idea of obtaining a random variable by integrating over the differential increments of the process in question makes sense. A good example of what one might be aiming for in all this would be a Fourier transform (initially defined formally, but then recast as a rigorous stochastic integral). Fourier analysis turns out to be particularly serviceable in the case of stationary processes, whether of a discrete parameter (chapter ten) or of a continuous parameter (chapter eleven). Here, in either case one seeks a spectral representation in terms of the covariance function. Given the spectral representation, one can perform linear operations with stationary processes as specified by a gain function—corresponding, in special cases, to ordinary differentiation or to integral averaging (that is, also referred to as filtering). The final chapter twelve covers linear least squares prediction of stationary (wide sense) processes, evidently an important application of the theory.
Doob’s exposition may be a little more leisurely and easier to follow than Feller’s (see our review here), but regrettably is virtually free of examples. It could also be faulted for being unilluminating: throughout, Doob’s proofs though highly technical do not appear to be very deep. The reason for this is that he stays at too high a level of generality. The derivations are clean and efficient, to be sure, but amount merely to a very general structural characterization of stochastic processes (akin to, in measure theory, the canonical decomposition of an arbitrary measure into its pure point, absolutely continuous and singular parts). In the absence of good examples, the danger remains that such general statements fail to acquire much concrete meaning in the mind of the student. What one will not find here are any refined results in the analysis of Brownian motion, such as the arcsine law or the law of the iterated logarithm.
Three stars: in the present work, there is altogether too much merely formal development without any exemplification in real-world use cases—an outcome which may be judged perhaps understandable given that, when Doob was writing, the theory had only just completed its initial stage of development (illustrating, thus, an unfortunate tendency in pure mathematics to run ahead with the formal elaboration in default of sufficient exploration of examples). The absence of homework exercises, moreover, hinders any deeper comprehension. This reviewer plans to go on next to another, decidedly recent work on stochastic processes by Rabi N. Bhattacharya and Edward C. Waymire, Stochastic Processes with Applications (SIAM Classics in Mathematics, 2009); he was pleasantly surprised by these authors’ elegant A Basic Course in Probability Theory (Springer-Verlag, second edition, 2016, our review here) and hopes that they have continued to deploy their crisp and clear modern style of exposition in their much longer comprehensive work on stochastic processes, which appears to be pitched to a reader in command, indeed, of a certain degree of mathematical maturity but seeking a good intuitive feel for the outlines of the subject more than unrelenting and painstaking rigor. So, be on the watch for a forthcoming review!
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