The volume under review bears a publication date 2001 and is based on a first year graduate course given from 1996 to 1999 at the Courant Institute of Mathematical Sciences. It has a clear affinity with the book by J. Jacod and P. E. Protter [Probability essentials, Springer, Berlin, 2000; MR1736066], which was based on a course at the University of Paris VI. But the differences are even more interesting than the similarities. If, in its 250 pages, the latter offers a ``lean one-semester course'' (as stated in its preface), then the present work must be said to offer, in its 165 pages, the bare bones of a year-long course. The difference in length is mainly due to the inclusion, in the former, of elementary material combining measure and probability, prior to the introduction of characteristic functions on p. 99, whereas the present work starts with a chapter of straight measure theory, and characteristic functions appear already on page 19. It is hard to escape the contrast in stereotypes between the suave Parisian (Probability essentials) and the brash New Yorker (Varadhan). In both books the characteristic functions are followed by some (classical) limit theorems for sums of independent random variables. But the present work goes farther in that it includes the basic limit theorem for the Lévy-Khinchin representation (Theorem 3.21). The proof is difficult, however, and neither book goes into class L or stable laws. Following independence comes dependence and conditional probability. Again, the necessary measure theory comes first for Varadhan, comprising a probability-free proof of the Radon-Nikodým theorem, while Jacod et al. use the L2-theory of random variables. Following this, the Varadhan book offers a thorough discussion of the Markov property in discrete time, including integer-parameter Markov chains with countable state space (none of which is given in Probability essentials). By way of examples there are five specializations: (1) simple random walk, (2) a queueing problem, (3) an urn scheme (namely Pólya's), (4) a branching process, and (5) a bilateral birth-and-death process. The asymptotic behaviors as the time n→∞ are worked out by methods which may appear ad hoc, in the absence of martingales. These last are treated immediately thereafter, including a first proof of the a.s. convergence of L1-bounded martingales without using stopping times. We then return to the Markov chains and the (Poisson) equation (II-I) V=0 (where II-I is the discrete time generator) whose solutions provide us with the needed martingale V(Xn). There are two more chapters: Chapter 6 on Stationary stochastic processes, and Chapter 7 on Dynamic control and filtering. The former is an excellent introduction to the subject, including the usual (Garsia) proof of the pointwise ergodic theorem. Considerable ingenuity has gone into the specialization of this to stationary Markov chains. There is also (Section 6.6) a thumbnail rendition of the spectral theory of stationary Gaussian processes. Since the final chapter is more in the nature of a pep talk than complete exposition, the capstone of the course, in our opinion, is the central limit theorem for martingale differences. This was the case for the earlier book also, but in place of Jacod and Protter's assumption E(|Xn|3/Fn−1)<K<∞, Varadhan assumes ergodic stationarity, thus avoiding any assumption on the third moments.
Reviewer: Knight, F. B. [form MathSciNet]