Although this book was first published in 1976, it has gained new significance and renewed interest among statisticians due to the developments of modern statistical techniques such as the bootstrap, the efficacy of which can be ascertained by asymptotic expansions. This also is the only book containing a detailed treatment of various refinements of the multivariate central limit theorem (CLT), including Berry-Essen-type error bounds for probabilities of general classes of functions and sets, and asymptotic expansions for both lattice and non-lattice distributions. With meticulous care, the authors develop necessary background on weak convergence theory, Fourier analysis, geometry of convex sets, and the relationship between lattice random vectors and discrete subgroups of Rk. The formalism developed in the book has been used in the extension of the theory by Goetze and Hipp to sums of weakly dependent random vectors. This edition of the book includes a new chapter that provides an application of Stein's method of approximation to the multivariate CLT.
Rabi N. Bhattacharya received his Ph.D. from the University of Chicago in 1967. He has held regular faculty positions at the University of California, Berkeley, Indiana University and the University of Arizona. He is a member of the American Mathematical Society and a Fellow of the Institute of Mathematical Statistics. He is a recipient of a Guggenheim Fellowship and an Alexander Von Humboldt Forschungspreis. Bhattacharya has co-authored a number of graduate texts and research monographs, including Stochastic Processes with Applications (with E. C. Waymire) and Random Dynamical Systems (with M. K. Majumdar). R. Ranga Rao received his Ph.D. from the Indian Statistical Institute in 1960. He has been on the faculty of the Department of Mathematics, University of Illinois, for more than forty years. He has held a number of visiting professorships, including several at the Tata Institute of Fundamental Research, India. He is a member of the American Mathematical Society.
Preface to the Classics Edition Preface Chapter 1: Weak Convergence of Probability Measures and Uniformity Classes Chapter 2: Fourier Transforms and Expansions of Characteristic Functions Chapter 3: Bounds for Errors of Normal Approximation Chapter 4: Asymptotic Expansions-Nonlattice Distributions Chapter 5: Asymptotic Expansions-Lattice Distributions Chapter 6: Two Recent Improvements Chapter 7: An Application of Stein's Method;Appendix A.1: Random Vectors and Independence;Appendix A.2: Functions of Bounded Variation and Distribution Functions Appendix A.3: Absolutely Continuous, Singular, and Discrete Probability Measures Appendix A.4: The Euler-MacLaurin Sum Formula for Functions of Several Variables References Index.
