## The Annals of Mathematical Statistics

### A Combinatorial Central Limit Theorem

Wassily Hoeffding

#### Abstract

Let $(Y_{n1}, \cdots, Y_{nn})$ be a random vector which takes on the $n!$ permutations of $(1, \cdots, n)$ with equal probabilities. Let $c_n(i, j), i,j = 1, \cdots, n,$ be $n^2$ real numbers. Sufficient conditions for the asymptotic normality of $S_n = \sum^n_{i=1} c_n(i, Y_{ni})$ are given (Theorem 3). For the special case $c_n(i,j) = a_n(i)b_n(j)$ a stronger version of a theorem of Wald, Wolfowitz and Noether is obtained (Theorem 4). A condition of Noether is simplified (Theorem 1).

#### Article information

Source
Ann. Math. Statist., Volume 22, Number 4 (1951), 558-566.

Dates
First available in Project Euclid: 28 April 2007

https://projecteuclid.org/euclid.aoms/1177729545

Digital Object Identifier
doi:10.1214/aoms/1177729545

Mathematical Reviews number (MathSciNet)
MR44058

Zentralblatt MATH identifier
0044.13702

JSTOR