Characteristic Functions, Moments, and the Central Limit Theorem

Abstract

In [2], Lindeberg conditions of order $\nu \geqq 2$ are defined and shown to be NSC for convergence of $\nu$th absolute moments in the Central Limit Theorem when $\nu = 2k, k = 2,3, \cdots$. Section 4 contains the extension of that result to the case of all $\nu > 2$, the proof depending on some of the theorems, given in Section 2, relating the existence of moments to the integrability of the characteristic function near the origin. The proofs of the results of Section 2 are deferred to Section 3 and depend, in turn, on known results listed in Section 1. Throughout, we use the notations $\mathfrak{RI}x, \mathfrak{Im}x$ for the real, imaginary (respectively) parts of $x$, and $\lbrack x \rbrack$ to mean the largest integer strictly less than $x$.