In this paper the relation between weak convergence of a sequence of measures and uniform convergence over certain classes of continuity sets (or uniform convergence of the integrals over certain classes of continuous functions) is studied. These results are applied to obtain laws of large numbers for random functions and generalizations of the Glivenko-Cantelli lemma.

The author is indebted to R R Bahadur for his encouragement and for valuable suggestions and discussions during the preparation of the paper.

The author is greatly indebted to R R Bahadur for encouragement and for valuable suggestions and discussions.

The purpose of this paper is to study a class of irreducible Banach space representations of a complex semi-simple Lie group G through an analysis of the associated representations of the Lie algebra of G.

The work described in this paper was done during 1963-65 when the authors were at the Indian Statistical Institute, Calcutta. At an early stage of the work, the last named author [Varadarajan] had the opportunity of detailed and highly stimulating discussions with Dr S R S Varadhan currently of the Courant Institute of Mathematical Sciences, New York. He would like to express his indebtedness to Dr Varadhan for these discussions which determined our entire approach to the whole circle of questions studied in the present work.

Ranga had once visited me at Berkeley on a sabbatical leave from Illinois. But after a couple of months at Berkeley, continuing problems with retina detachment forced him to return to Urbana. For collaborating on the book, I would, therefore, visit him in Urbana for weeks at a stretch with my wife and young daughter. We would always stay at Ranga's house and enjoy his hospitality and that of his wife Shantha. Ranga and Shantha have been among the most generous and gentle souls I know. Although Ranga was no longer seriously interested in probability, I learnt from him some important mathematics related to the subject matter of the book. It is a pity that such a brilliant mathematician as Ranga Rao could not fulfil his exceptional potential because of his extremely poor eye sight! The book that we wrote was well received.

This monograph presents in a unified way various refinements of the classical central limit theorem for independent random vectors and includes recent research on the subject. Most of the multidimensional results in this area are fairly recent, and significant advances over the last 15 years have led to a fresh outlook. The increasing demands of application (e.g., to the large sample theory of statistics) indicate that the present generality is useful. It is rather fortunate that in our context precision and generality go hand in hand.

Apart from some material that most students in probability and statistics encounter during the first year of their graduate studies, this book is essentially self-contained. It is unavoidable that lengthy computations frequently appear in the text. We hope that in addition to making it easier for someone to check the veracity of a particular result of interest, the detailed computations will also be helpful in estimations of constants that appear in various error bounds in the text. To facilitate comprehension each chapter begins with a brief indication of the nature of the problem treated and its solution. Notes at the end of each chapter provide some history and references and, occasionally, additional facts. There is also an Appendix devoted partly to some elementary notions in probability and partly to some auxiliary results used in the book.

We have not discussed many topics closely related to the subject matter (not to mention applications). Some of these topics are "large deviation," extension of the results of this monograph to the dependence case, and rates of convergence for the invariance principle. It would take another book of comparable size to cover these topics adequately.

We take this opportunity to thank Professors Raghu Raj Bahadur and Patrick Billingsley for encouraging us to write this book and giving us advice.

This is the first book devoted to a study of rates of convergence to normality for normed sums of independent random vectors. The results almost all deal with the multidimensional case and hence are often not quite as precise as can be achieved in the one-dimensional case. Furthermore, necessity conditions are not generally available. A limited discussion of the one-dimensional and a comprehensive discussion of the multidimensional literature is provided as is a considerable amount of new material.
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There are several excellent monographs on limit theorems for sums of independent random variables and rates of convergence in these limit theorems, but until now there wasn't any monograph concerning these problems in k-dimensional case. The book under reviewing presents the first attempt to put in one place recent refinements of the classical central limit theorem for sums of independent random vectors (bounds for the remainder term and asymptotic expansions). The authors, whose contribution to this topic is well-known, carefully and consistently present the most general results, obtained by the method of characteristic functions. The book is self-contained, but it is not very easy to read it, since there are many lengthy computations. Fortunately, at the beginning of each chapter there is a short discussion of the problem treated in the chapter and its solution. From this discussion the reader can get a clear picture about results and main ideas involved in proofs (smoothing, truncation).

It is with great pleasure that the authors welcome the publication by SIAM of the edited reprint of 'Normal Approximation and Asymptotic Expansions'. The original edition was published in 1976 by Wiley, followed by a Russian translation in 1982 and an edited version with a new chapter by Krieger in 1986. The book has been out of print for nearly twenty years. Statistical applications such as "higher order" comparisons of efficiency, and the evaluation of the improvement over the classical central limit theorem due to the widely popular and important bootstrap methodology of Efron, have led to a renewed interest in the subject matter of the book. We also note with a measure of happiness that the theory of asymptotic expansions for sums of weakly dependent random variables/vectors due to Götze and Hipp made use of some of the formalism and estimation in our book. We have controlled an initial impulse to present this theory, as it would make the book substantially increase in size and take us much time to ready it for publication.

Keeping to independence, however, a short new chapter is added on Götze's application to the multivariate CLT of an ingenious method of Stein. The exposition, and a somewhat modified treatment presented here of the rather difficult original paper, resulted from a collaboration between one of the authors and Professor Susan Holmes.