## The Annals of Mathematical Statistics

### A Converse to a Combinatorial Limit Theorem

J. Robinson

#### 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.

#### Article information

Source
Ann. Math. Statist., Volume 43, Number 6 (1972), 2053-2057.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177690884

Digital Object Identifier
doi:10.1214/aoms/1177690884

Mathematical Reviews number (MathSciNet)
MR370704

Zentralblatt MATH identifier
0253.60027

JSTOR