Sums and Gaussian vectors
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 Gaussian r.e. in a separable Banach space, asymptotics of probabilities of small balls. Chapter 3 is devoted to the inequalities for sums of independent random vectors including exponential bounds, Hoffmann-Jørgensen, Nagaev-Fuk and Burkholder's inequalities. In Chapters 4 and 6 large deviations are treated, that is, estimates and asymptotic behavior of probabilities \(p_n (B, \varepsilon_n) = P \{\varepsilon_n \sum^n_{i = 1} \xi_i \in B\}\) are investigated. Here a set \(B\) and the normalizing factors \(\varepsilon_n > 0\) are chosen in such a way that this probability is infinitesimal. Chapter 4 is devoted to the so-called rough asymptotics of large deviations, i.e., to the asymptotics of \(\log p_n (B, \varepsilon_n )\), and in Chapter 6 fine asymptotics of moderate deviations (``moderate'' means that \(\varepsilon_n = (r \sqrt n)^{-1}\) with \(r = r(n) \to \infty\) and \(r = o (\sqrt n))\) is considered. Finally, Chapter 5 is devoted to Gaussian approximation for distribution of sums, that is, to the central limit theorem and its refinements such as Bergström and Edgeworth type asymptotic expansions. This chapter probably is the most technically complicated one in the book and as it is written by the author ``the problem of approximation is considered here from a rather special viewpoint. Attention is concentrated on some aspects of the construction which are of interest mainly in infinite dimensions and no attempt is made to include all known facts.'' At the end of the book there are bibliographic commentary and a comprehensive reference list.
Bergström and Edgeworth type asymptotic expansions
scientific article; zbMATH DE number 825910