The goal of this paper is to prove several estimates of the rate of convergence in the multi-dimensional central limit theorem in the uniform metric. Additionally, estimates will be derived for the rate of convergence in variation. These estimates are mainly derived from properties of the metrics. Formulation and Discussion of the ResultsConsider a sequence of independent identically distributed r.v.'s X1, X2, with values in Euclidean space R k. Let EX1 0, EIXIa< 0. Here[. [denotes the norm in R k. Below all the norms which we encounter will be denoted by I'1 should cause no misunderstandings. Denote by P and Pn the distributions of X1 and (X +. + Xn)n -1/2, respectively. We shall assume for simplicity that the covariance operator of the distribution P is the identity operator on R k. Denote by the normal law with zero mean and covariance operator tr2/, where ! is the identity operator in R k, and denote by . We shall use the following metrics: p(P, O)=sup{le(A)-O(A)l: A (}, where denotes the set of all convex Borel subsets of R k (we shall call p a uniform metric), as well as (P, O)= f IP-OI (dx), Var where s is the class of all real functions on R such that the integer rn and the number 0 < a =< 1 are such that m + a s, and f(') denotes the rnth FrOchet derivative of f, and by the norm of the derivative we mean the usual norm of a multilinear functional (see [2]-[5] for more details on the metrics (s).

v. SENATOV (Represented by V. M. Zolotarev) (Translated by V. V. Sazonov) Consider independent and identically distributed random variables X1, X2, with values in the Hilbert space 12. Assume that E X1 0 and E IX 2 < , where I" is the norm in 12. Denote by P the distribution of X1 and by Pn the distribution of Zn (X /... / Xn)n-/2. Let a 2, a22,.., be the eigenvalues of the covariance operator B of P, which we may assume, without loss of generality, to be ordered: a 2 _> a _> and let Af be the normal law with mean zero and covariance operator B. Furthermore, let SR(a) (x: Ixal <_ R} be the ball with radius R and center at a in/2; Cs Cs(P, Af) where Cs(', ") are the ideal metrics introduced by Zolotarev; / tt(P, Af), where tt(., .) is the uniform distance with respect to 2 be the trace of B. the class of convex Borel sets in /2; s f IXl P(dx) < cx; a 2 -]i=1 tri The distribution P may be degenerate in 12. The changes needed in this case in the estimates presented below are quite obvious. We are interested in bounds for ", o) Besides An we shall consider also the distances and pR(Pn, Jf) sup {An(P, r, a): a e/2, r <_ R} Put pd(pn, Af)= sup {An(P, r, a): lal _< d, _> 0}, d>O.Ai Ai(n) a2nl/2 -nl/-----, i-e e(R, n) (R33n-1/2) 1/6 i--1, 2..., 5(R, n) (Rln-1/2)1/2, and fix some (large enough) number -> 0; c, ci (ci _> 1) will denote constants and c(a, b,...) quantities depending only on a, b, THEOREM 1. For any R

В статье приводятся явные оценки точности, которые гарантируют асимптотические разложения в центральной предельной теореме. Ключевые слова и фразы: центральная предельная теорема, асим птотические разложения, точность аппроксимации. Московский государственный университет им. М. В. Ломоносова, механикоматематический факультет, кафедра теории вероятностей,

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