Open Access
April, 1970 Characteristic Functions, Moments, and the Central Limit Theorem
B. M. Brown
Ann. Math. Statist. 41(2): 658-664 (April, 1970). DOI: 10.1214/aoms/1177697109


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 $x$.


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B. M. Brown. "Characteristic Functions, Moments, and the Central Limit Theorem." Ann. Math. Statist. 41 (2) 658 - 664, April, 1970.


Published: April, 1970
First available in Project Euclid: 27 April 2007

zbMATH: 0196.21204
MathSciNet: MR261672
Digital Object Identifier: 10.1214/aoms/1177697109

Rights: Copyright © 1970 Institute of Mathematical Statistics

Vol.41 • No. 2 • April, 1970
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