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.
Citation
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
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