Publications results for "Reviewer=(aldous) AND Publication Type=(Books)"
MR0624435 (83a:60001)
Hall, P.; Heyde, C. C.
Martingale limit theory and its application.
Probability and Mathematical Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. xii+308 pp. ISBN: 0-12-319350-8
60-02 (60B12 60F05 60G42)
More links
PDF Doc Del Clipboard Journal Article Make Link

In retrospect, it is surprising that not until around 1970 was it realised that the crucial condition for the central limit theorem (CLT) for martingales $S_n=\sum_1^nX_i$ was $V_n^2/E(V_n^2)\rightarrow 1$, where $V_n^2$ denotes the conditional variance $\sum_1^nE(X_i^2|\scr F_{i-1})$. During the 1970s many extensions and refinements of this idea appeared in the literature, and the need arose for a definitive reference work on "weak" martingale limit theorems, as a complement to J. Neveu's book [ Discrete-parameter martingales, English translation, North-Holland, Amsterdam, 1975; MR0402915 (53 \#6729)] which treats only "strong" limit theorems. This book addresses that need, in the discrete-time setting.

The book starts with a brisk treatment of the a.s. convergence theory, laws of large numbers and square function inequalities. Chapter 3, on the CLT, is the heart of the book. McLeish's slick characteristic function proof of the basic result is used, and there follows discussion of convergence to mixtures of normals, convergence of moments, rates of convergence and reverse martingales. In Chapter 4 the Skorokhod embedding technique is used to treat weak convergence of martingales to Brownian motion, and rates of convergence are again given. Next the functional form of the law of the iterated logarithm (LIL) is discussed. In Chapter 5 the authors discuss limit theorems for stationary processes. Here Gordin's technique of approximating stationary sequences by martingales is developed to establish a CLT and functional CLT and LIL; also Ibragimov's CLT under the hypothesis of vanishing maximal correlation is given. In Chapter 6 the authors describe applications in statistics, specifically to the estimation of parameters in stochastic processes. Martingales arise naturally in studying the asymptotic properties of the MLE, but the book also gives applications in conditional least squares estimation, and in estimating autocorrelations in time series. Chapter 7, aptly entitled "Miscellaneous applications", includes de Finetti's theorem for exchangeable sequences, Kingman's subadditive ergodic theorem, stochastic approximation methods and several other topics.

The reviewer liked the book very much. It gives a good view of the "state of the art" in the theory, and a wide sample of applications; it should succeed in becoming the standard reference work in this area. The reviewer's only criticisms are of omissions, mainly of the lack of discussion of the continuous-time case. Here it is known that essentially the same results hold, with the role of the conditional variance $V_n^2$ being played by $\langle M,M\rangle_t$. A chapter on this would have made the book substantially more useful as a reference work. Also, for stationary sequences there are strong approximation techniques which give stronger conclusions under stronger hypotheses: more discussion of the range of applicability of the various hypotheses used on stationary sequences would have been helpful. Finally, work by I. S. Helland [Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 251--265; MR0576886 (81j:60045)] on weak convergence to Brownian motion appeared just too late to be noted in the bibliography, which seems otherwise complete as far as the main "weak limit theorem" topics are concerned.

Reviewed by David J. Aldous
American Mathematical Society
American Mathematical Society
201 Charles Street
Providence, RI 02904-2294