Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons. X, 534 p. £ 49.10 (1986).

In this book the authors’ principal intention is to provide tools for establishing the weak convergence of sequences of Markov processes to (typically) diffusion limits. The usual way of doing this, analogous to the use of characteristic function methods in the classical CLT, is to find some property of the processes whose convergence is simpler to verify, but nonetheless implies convergence of the processes themselves. If such a method is to work, it is clear that the property chosen must characterize the Markov processes. Conversely, for tight sequences of measures, the usual subsequence argument shows that the fact that the property chosen characterizes the processes is pretty well enough to make the method work. This has led the authors to extend the scope of the book, and to treat the problem of characterization as a subject in its own right.
There are three main themes. The first, {\it H. F. Trotter}’s [Pac. J. Math. 8, 887-919 (1958;

Zbl 0099.103)] approach, is based on operator semigroup theory, and the property considered is the generator of the semigroup determining the Markov process. This leads to conditions for convergence involving the convergence of generators, together with the requirement that the limit of the generators itself generates a strongly continuous contraction semigroup, the study of the latter condition being the corresponding characterization problem.
The second theme is the martingale theory of {\it D. W. Stroock} and {\it S. R. S. Varadhan} [Multidimensional diffusion processes. (1979;

Zbl 0426.60069)], and the property considered is again the generator, this time arising from the local characteristics of the martingales characterizing the Markov processes. The corresponding characterization problem is then the uniqueness of solutions of the martingale problem associated with the limit of the generators.
The third theme, exploiting the idea of operational times as developed by the second author [Stochastic nonlinear systems in physics, chemistry and biology, Proc. Workshop, Bielefeld 1980, Springer Ser. Synerg. 8, 22-35 (1981;

Zbl 0461.60090)] is to represent the Markov processes as the solutions of certain implicit equations involving standard processes such as the Poisson process. If the equations converge in an appropriate sense, so do the Markov processes, provided always that the limiting equation has a unique solution, this leading to the corresponding characterization problem.
The book is written as a self-contained reference text for those already acquainted with the elements of probability, measure and functional analysis. As such, it succeeds admirably, providing a handy source of useful and detailed information on the basic theory required for each of the three techniques, in addition to a careful and exhaustive treatment of the subject matter proper. In order to achieve this, a large amount of material has had to be squeezed into its 500 pages. Thus the opening chapters on Operator semigroups, Stochastic processes and martingales and Convergence of probability measures include the Hille-Yosida theorem, Trotter’s product formula, the Doob-Meyer decomposition, the Skorohod representation and Aldous’ condition for tightness (in $D\sb E[0,\infty))$, all within 150 pages. Not surprisingly, the representation is somewhat terse. As a measure of compensation, there are notes on the literature and a selection of problems at the end of each chapter, designed to help in the construction of a graduate course based on the book, and the techniques are illustrated in the last four chapters in proving limit theorems for branching processes, genetical models, population processes and random evolutions.
There is no question but that space should immediately be reserved for the book on the library shelf. Those who aspire to mastery of the contents should also reserve a large number of long winter evenings.