268 Book Reviews

martingale convergence theorem (for square integrable martingales) and the optional stopping theorem are proved here, with many examples. The remaining four chapters are on more general random processes; here we have stationary processes, diffusions, renewal theory and queueing theory. My feeling is that these chapters are less successful than the earlier ones. For one thing, there are already some well-established textbooks on this material at about the same level, and I feel that Grimmett and Stirzaker do not have much new to offer. Furthermore, the material on "Stationarity and Diffusion" (taken together in a single chapter!) is really too sketchy. The section on stationary processes includes the spectral and ergodic theorems, but with no discussion of filtering or prediction it must be hard to see why these concepts are useful. Diffusions are introduced through the Wiener and Ornstein-Uhlenbeck processes, with only a very brief discussion of general diffusion processes. Chapter 10, on renewal theory, is very similar to the treatment in Karlin and Taylor [4], while the final chapter, on queues, is mostly well-established material. One of the best features of the book is its problems, many of which are imaginative and challenging. Sketched solutions are given at the back of the book. There are also some excellent examples in the text, though perhaps too many on the hackneyed themes of coins, cards and dice, especially in the early chapters. In later chapters there are many examples taken from biology, though very few from physics or engineering, which is a pity. A small but not insignificant quibble is that the references are poor. There is a useful bibliography but many of the examples and problems are given without reference to the original sources. A case in point is Etemadi's proof—Etemadi's name is mentioned but the exact reference is not. To this the authors might reply that they are only following the tradition established by Feller, but I feel it is a bad tradition. I believe this book will be most valuable to postgraduates and research workers in mathematics or statistics needing a quick but thorough introduction to probability, and to advanced undergraduates specialising in probability. I would hesitate to recommend it as an introductory text to all undergraduates, as the pace is fast and a definite level of maturity is required of the reader. In conclusion, it goes a long way towards filling the gap referred to in my opening paragraph, but perhaps does not close it completely.