On Lindeberg-Feller Limit Theorem

In the Lindeberg-Feller theorem, the Lindeberg condition is present. The fulfillment of this condition must be checked for any ε > 0. We formulae a new condition in terms of some generalization of moments of order 2 + α, which does not depend on ε, and show that this condition is equivalent to the Lindeberg condition, and if this condition is valid for some α > 0 then it is valid for any α > 0. In the non-classical setting (in the absence of conditions of a uniform infinitely smallness) V.I. Rotar formulated an analogue of the Lindeberg condition in terms of the second pseudo-moments. The paper presents the same modification of Rotar`s condition, which does not depend on ε. In addition, we discuss variants of the simple proofs of theorems of Lindeberg-Feller and Rotar and some related inequalities.