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Book review

Author(s) Paulauskas, V.
Rackauskas, A.
Title Approximation theory in the central limit theorem
Publisher Kluwer Academic Publishers
Year of publication 1989
Reviewed by Vasile Postolica

The present book is a revised translation of the original work in Russian language published by Mokslas, Vilnius 1987. It deals with problems concerning with the estimates of the rate of convergence in the central limit theorem in Banach spaces (mainly, the estimates of convergence rate on sets with a non-smooth boundary in general Banach spaces and in different metrics), being based on the results obtained by Vilnius mathematicians in this field.

After a very clear introduction, the authors consider in the first chapter the definitions, concepts and results in the theory of the distributions in Banach spaces which are necessary in the following chapters. The construction of smooth functions "well approximating" the indicator functions of given sets in Banach spaces is tackled in the second chapter. The main results in the third chapter are the central limit theorems in the spaces C(S) and c0 obtained by the use of type-2 operators. The main aim of the fourth chapter is to give emphasis to the essential difference in finite-dimensional and infinite-dimensional spaces for the Gaussian measures of epsilon-strips for convex sets. The estimates of the rate of convergence in the central limit theorem in Banach spaces are presented in the fifth chapter, the last but the main part of the book. Finally, we find important bibliographical remarks and references. Since the textbook is written clearly with many valuable considerations for numerical methods, we recommend the book to all scientists interested to analyse the accuracy in approximating the distribution of sums of independent identically distributed random elements in Banach spaces by Gaussian distributions.