548 Book Reviews

Though one can easily build a diffusion in natural scale with a given speed measure m from standard Brownian motion and its local time, the problem is to show that this gives every diffusion in natural scale; Freedman shows that the speed measure determines the generator which determines the diffusion, thus disposing of this point. However, it is quite a lengthy business. From this analysis of one-dimensional diffusions springs the miraculous Feller-McKean chain and others with similar weird properties. The final section presents some examples of one-dimensional continuous processes which show that Freedman was well aware of the limitations of the theory as it then stood. What prevents the first part of the book from being just another account of Brownian motion is Freedman's choice of examples and applications. The results on quadratic variation of the Brownian sample path and on Strassen's law of the interated logarithm are of great interest, but rarely get such a thorough treatment. The accounts of the Skorokhod representation, martingale central limit theorems, and Wald's identity have all been surpassed by subsequent developments, but are still quite readable introductions. Perhaps it is unfair to judge a book with hindsight; but, nonetheless, the Freedman trilogy has been republished in 1983, so must be judged by the standards of 1983. When it first appeared, the trilogy was excellent, and quickly acquired admirers: after twelve years, it is only good; but it is still good.