Sums and Gaussian vectors
type: Item
instance of: scholarly article
title: Sums and Gaussian vectors
zbMATH Open document ID: 0846.60003
published in: Lecture Notes in Mathematics
publication date: 1995-12-13T00:00:00Z
review text: The book written by one of the leading Russian probabilists working on limit theorems of probability theory is devoted to the central limit theorem for sums of i.i.d. random elements with values in spaces with ``very high'' or infinite dimensions. The author's attention is concentrated on classical topics of limit theorems -- asymptotic expressions for large deviation probabilities, inequalities and normal approximation including asymptotic expansions for distributions of sums. Although these topics were given considerable attention during the last decades, still we can not say that the picture of the central limit theorem with refinements in general Banach spaces is clear and well understood. Therefore to each author writing a book in this area (here may be it is appropriate to mention that the number of such authors is not very big) there is a dilemma to choose between the transparency and completeness of exposition. How the author solved this dilemma one can get an idea from the following sentences from the preface of the book: ``The compromise attempted here is to provide a reasonably detailed view of the ideas that have already gained a firm hold, to make the treatment as unified as possible, and sacrifice some of the details that do not fit into the scheme or tend to inflate the text beyond reasonable limits. The price is that such a selection inevitably results biased, and one of the sacrifices was the refined CLT itself.'' Now we shall describe shortly the content of the book. The first chapter is devoted to Gaussian measures in Euclidean spaces and their basic properties, such as isoperimetric inequality and smoothness properties of some convex functionals of Gaussian random elements. Chapter 2 is devoted to seminorms of Gaussian vectors in infinite-dimensional spaces, and the following results are presented: existence of exponential moments of a seminorm of a Gaussian random element (r.e.), absolute continuity and other smoothness properties of a norm of G…
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 825910
zbMATH Keywords: Gaussian approximation
zbMATH Keywords: large deviations
zbMATH Keywords: inequalities
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: seminorms of Gaussian vectors
zbMATH Keywords: asymptotics of moderate deviations
zbMATH Keywords: Bergström and Edgeworth type asymptotic expansions
author: V. V. Yurinsky
reviewed by: V. Paulauskas
MaRDI profile type: Publication
DOI: 10.1007/BFB0092599
description: scientific article; zbMATH DE number 825910