By Erkus E., Duman O.

During this paper, utilizing the idea that ofA-statistical convergence that is a regular(non-matrix) summability strategy, we receive a common Korovkin kind approximation theorem which matters the matter of approximating a functionality f through a series {Lnf } of confident linear operators.

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21 ) vn. 22) valid for all E > 0 and I = 1, 2, .... l via IIxl1 2 ~ L~=l (x, es )2. l concentration inequalities of this kind have been proved by Esseen (1968), Paulauskas (1973), Gotze (1979), etc. , Siegel (1981), Bentkus (1985b), etc. l can be obtained as follows. e. 5. Let us estimate h Put G(x) = exp(itllxI1 2). Then G E Coo and G'"(x)h 3 = -8t 2 G(x)[it(x, h)3 The estimate h = O(n-l/2) + (x, h)(h, h)]. 23) II. The Accuracy of Gaussian Approximation in Banach Spaces IEG(Sn) - EG(Y)I = 0(ltln- 1/ 2 ).

4, 810-820. Engl. : Theory Probabl. Appl. 20, Mno. 4, 794-804. Zbl. V. (1978a): On a lower bound for the remainder in the central limit theorem. 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.

4 are useful for the case of non-smooth functions F. The last section of the chapter discusses the results concerning the convergence rate in the CLT, estimated by means of the Prokhorov and bounded Lipschitz (BL) metrics. The second chapter is devoted to asymptotic expansions. 2 we consider asymptotic expansions for the expectation Ef(Sn) with f : B - IR a sufficiently smooth function or f a function having isolated points of nondifferentiability. The expectation EllSn lIP, p > 0, with a sufficiently smooth norm-function, is a typical example.

### A -Statistical extension of the Korovkin type approximation theorem by Erkus E., Duman O.

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