Rabi N. Bhattacharya

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.

This book develops systematically and rigorously, yet in an expository and lively manner, the evolution of general random processes and their large time properties such as transience, recurrence, and convergence to steady states. The emphasis is on the most important classes of these processes from the viewpoint of theory as well as applications, namely, Markov processes. It features very broad coverage of the most applicable aspects of stochastic processes, including sufficient material for self-contained courses on random walk in one and multiple dimensions; Markov chains in discrete and continuous times, including birth-death processes; Brownian motion and diffusions; stochastic optimization; and stochastic differential equations. Most results are presented with complete proofs, while some very technical matters are relegated to a Theoretical Complements section at the end of each chapter in order not to impede the flow of the material. Chapter Applications, as well as numerous extensively worked examples, illustrate important applications of the subject to various fields of science, engineering, economics, and applied mathematics. The essentials of measure theoretic probability are included in an appendix to complete some of the more technical aspects of the text.