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Approximation theory in the central limit theorem. Exact results in Banach spaces. Transl. from the Russian by B. Svecevičius and V. Paulauskas. (English) Zbl 0715.60023
Mathematics and Its Applications. Soviet Series 32. Dordrecht etc.: Kluwer Academic Publishers (ISBN 90-277-2825-9). xviii, 156 p. (1989).
The monograph is devoted to the rate of convergence in the central limit theorem in Banach spaces. The general setup consists of a sequence of i.i.d. vectors \(X_ j\) taking values in a separable Banach space B which have mean zero such that there exists a Gaussian random vector Y with the same covariance operator. Assuming that the third absolute moment of \(X_ j\) exists the authors investigate the rate of convergence of the distribution of a normalized sum \(\rho_ n\triangleq n^{-1/2}(X_ 1+...+X_ n)\) to the distribution of Y. As measures of discrepancy they consider the Kolmogorov supremum norm for a class of sets \({\mathcal A}\), the Prokhorov and the Lipshitz metric. The main emphasis lies on the first distance where the authors use Lindeberg’s method together with induction to prove bounds of order \(O(n^{-1/6})\) for certain classes \({\mathcal A}\) (e.g. balls) such that: 1) indicator functions of sets from \({\mathcal A}\) have a smooth approximation (Chapter 2) and 2) the Gaussian measure of the \(\epsilon\)-boundary can be estimated in terms of \(\epsilon\) (Chapter 4).
In Chapter 5 the authors give examples where these rates are unimprovable. Furthermore, dimensional approximation methods (finite and infinite) and methods of characteristic functions for balls in Hilbert space due to Götze and Yurinskii are discussed. Finally, the last section of Chapter 5 is devoted to proving rates of order \(O(n^{-1/8})\) for the Prokhorov metric resp. \(O(n^{-1/6})\) for the bounded Lipshitz metric.
The monograph contains results on the rates of convergence in the CLT in the i.i.d. case up to 1986 and provides a rather complete picture of this topic up to this date.
Reviewer: F.Götze

60F05 Central limit and other weak theorems
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F17 Functional limit theorems; invariance principles
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