On Lindeberg–Feller Limit Theorem

Abstract

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 + \(\alpha \), 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 \(\alpha > 0\) then it is valid for any \(\alpha \) > 0. In the nonclassical 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-momens. 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.