ИСТИНА

We discuss the history of the Central Limit Theorem in a classical and nonclassical setting and give in a sense a simple proof of Lindeberg-Feller theorem. Lindeberg characteristic can be written in the following form: L_n(\varepsilon)=\sum E(X_{nk}^2 I(|X_{nk}|>\varepsilon)), where X_{nk}, 1\leq k\leq k_n are the normalized terms of the sum. For the case of nonclassical setting Rotar introduced an analogue of Lindeberg characteristic where instead of the second ε-tail moment he uses the second ε-tail absolute difference pseudo-moment. We present the following modification of the Lindeberg characteristic: L_n^b=\sum E(X_{nk}^2 b(X_{nk})), where b=b(x)\in B, B - is very broad class of functions (in particular \max[x^{\alpha},1]\in B for any α>0). We prove that the following three conditions are equivalent: a) L_n^{b_0}\to 0 for some b_0 \in B, b) L_n(\varepsilon)\to 0 for any \varepsilon>0, c) L_n^b\to 0 for any b\in B. We introduce also a similar modification of Rotar characteristic and proof a similar statement for nonclassical setting.