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June, 1950 On the Asymptotic Distribution of the Sum of Powers of Unit Frequency Differences
Bradford F. Kimball
Ann. Math. Statist. 21(2): 263-271 (June, 1950). DOI: 10.1214/aoms/1177729843

Abstract

Since the "unit" frequency differences (see (2.2) below) are dependent, the usual methods for establishing the normal character of the asymptotic distribution of the sum of random variables fail. However, the essential character of the distribution is disclosed by the integral functional relationship (3.6). From this it is possible to show that for large samples the distribution approximates "stability" in the normal sense ([2] and Lemma 2). Using the condition that the third logarithmic derivative of the characteristic function is uniformly bounded for all $n$ on a neighborhood of $t = 0$ one can prove that the asymptotic distribution exists and is normal.

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Bradford F. Kimball. "On the Asymptotic Distribution of the Sum of Powers of Unit Frequency Differences." Ann. Math. Statist. 21 (2) 263 - 271, June, 1950. https://doi.org/10.1214/aoms/1177729843

Information

Published: June, 1950
First available in Project Euclid: 28 April 2007

zbMATH: 0038.09403
MathSciNet: MR34987
Digital Object Identifier: 10.1214/aoms/1177729843

Rights: Copyright © 1950 Institute of Mathematical Statistics

Vol.21 • No. 2 • June, 1950
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