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3, 403-410. Engl. : Math. Notes 24, No. 3-4, 715-719. Zbl. V. (1978b): On the exactness of an estimate of the remainder term in the central limit theorem. Teor. Veroyatn. Primen. 4, 744-761. Engl. : Theory Probabl. Appl. 4, 712-730. Zbl. 1. (1984): Rate of convergence in an invariance principle for nonidentically distributed variables with exponential moments. Tru. Inst. Mat. 3, 4-49. Engl. : Advances in Pribability Theory, Transl. for Math. Engl. 2-73 (1986). Zbl. 1. (1985): Estimates in an invariance principle, Tru.

D. 's such that EX = 0 Y, Y 1 , Y z , ... d. e. Ef(X)g(X) = Ef(Y)g(Y) for all f,g E B*. B,F B* Sn = Sn(X) = n-1/Z(X1 + ... + Xn). C(X) the distribution of X. v = C(Y) the distribution of Y. cov X the covariance operator of X. By definition cov X B* --t B, covX(f) = Ef(X)X for all f E B*. (3s = EIIXII", s E ~ the s-th moment of X. Vs = vs(C(X),C(Y)) = IB IlxWIC(X) - C(Y)I(dx) the s-th pseudomoment. IJ. L - v. XA ( t) the indicator function of a set A. f(s)(x) the s-th Frechet derivative of the function f at the point x.

S such that EX = EY = 0, cov X = cov Y and Y is Gaussian. Furthermore, Sn = n- 1 / 2 (X1 + ... d. Briefly, Lindeberg's approach is as follows. , we have to estimate the quantity J-Ln(A) - v(A) = 1 XA(X)(J-Ln - v)(dx). The first step is to replace the discontinuous indicator function XA by a sufficiently smooth function say 9 = gA,g, which coincides with XA everywhere with the exception of an c-neighborhood (8A)" = {x E B : infYE8A IIx - yll < c} of the boundary 8A of the set A. After this substitution (usually called the smoothing lemma) two terms are to be estimated: the integral g(x)(J-Ln - v)(dx) and the quantity v«8A),,).

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