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December, 1972 A Converse to a Combinatorial Limit Theorem
J. Robinson
Ann. Math. Statist. 43(6): 2053-2057 (December, 1972). DOI: 10.1214/aoms/1177690884

Abstract

Let $a_n(i), b_n(i), i = 1, \cdots, n$, be $2n$ numbers defined for every $n$ and let $\bar{A}(k) = \sum^n_{i=1} |a_n(i)|^k$ and $\bar{B}(k) = \sum^n_{i=1}|b_n(i)|^k$. Let $(I_{n1}, \cdots, I_{nn})$ be a random permutation of $(1, \cdots, n)$ and let $S_n = \sum^n_{i=1} b_n(i)a_n(I_{ni})$. If $\bar{A}(k)/\lbrack\bar{A}(2)\rbrack^{\frac{1}{2}k} \rightarrow 0\quad \text{and}\quad \bar{B}(k)/\lbrack\bar{B}(2)\rbrack^{\frac{1}{2}k} \rightarrow 0.$ then it is known that the condition of Hoeffding, $n^{\frac{1}{2}k-1} \bar{A}(k)\bar{B}(k)/\lbrack\bar{A}(2) \bar{B}(2)\rbrack^{\frac{1}{2}k} \rightarrow 0,\quad k = 3,4, \cdots,$ is sufficient for the standardized moments of $S_n$ to tend to the moments of a standard normal variate. It is shown here that these conditions are also necessary. The relationship of these conditions to the Liapounov conditions is pointed out.

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J. Robinson. "A Converse to a Combinatorial Limit Theorem." Ann. Math. Statist. 43 (6) 2053 - 2057, December, 1972. https://doi.org/10.1214/aoms/1177690884

Information

Published: December, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0253.60027
MathSciNet: MR370704
Digital Object Identifier: 10.1214/aoms/1177690884

Rights: Copyright © 1972 Institute of Mathematical Statistics

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Vol.43 • No. 6 • December, 1972
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