Limit Theorems of Probability Theory

This book consists of five parts written by different authors devoted to various problems dealing with probability limit theorems. The first part, "Classical-Type Limit Theorems for Sums ofIndependent Random Variables" (V.v. Petrov), presents a number of classical limit theorems for sums of independent random variables as well as newer related results. The presentation dwells on three basic topics: the central limit theorem, laws of large numbers and the law of the iterated logarithm for sequences of real-valued random variables. The second part, "The Accuracy of Gaussian Approximation in Banach Spaces" (V. Bentkus, F. G6tze, V. Paulauskas and A. Rackauskas), reviews various results and methods used to estimate the convergence rate in the central limit theorem and to construct asymptotic expansions in infinite-dimensional spaces. The authors con fine themselves to independent and identically distributed random variables. They do not strive to be exhaustive or to obtain the most general results; their aim is merely to point out the differences from the finite-dimensional case and to explain certain new phenomena related to the more complex structure of Banach spaces. Also reflected here is the growing tendency in recent years to apply results obtained for Banach spaces to asymptotic problems of statistics.

The book covers aspects of limit theorems by a well-known group of researchers.

Zusammenfassung

"The book is a very fine collection of five authoritative essays on five main directions in asymptotic probability, each giving a state-of-the-art overview as seen from Vilnius ... in 1991, when the original Russian edition was published in Moscow. Of course, any insider appreciates the relevance, indeed, the importance of the Vilnius view. The nine-year delay with this English translation (a superb job, done by B.Seckler) is unessential: while all five fields continued to develop and grow in these nine years, the essayists make honest attempts to explore the very essence of their fields, and mostly with success. Thus, most later developments of note would easily fit into the framework described in each of the five directions. ... People doing research in any of the areas above will find the volume extremely useful as a reference work that, in particular, makes hundreds of papers from Russian periodicals available for the first time. It will be practically indispensable for a graduate student beginning to do research in asymptotic probability."

Sándor Csörgö, Acta Scientiarum Mathematicarum 2002, Vol. 68, pp. 1013-1014

"... This book should be at hand for any specialist on asymptotic methods in probability and statistics, and can also serve as a basic reference for graduate and Ph.D. students of stochastics."

European Mathematical Society, March 2002

"The monograph is written at the research level and will be a useful reference for graduate students and teachers."

Journal of the American Statistical Association

Inhalt
I. Classical-Type Limit Theorems for Sums of Independent Random Variables.- II. The Accuracy of Gaussian Approximation in Banach Spaces.- III. Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal Distribution.- IV. Refinements of the Central Limit Theorem for Homogeneous Markov Chains.- V. Limit Theorems on Large Deviations.- Name Index.