On the Asymptotic Normality of Certain Rank Order Statistics

Abstract

Let $(R_1, \cdots, R_N)$ be a random vector which takes on each of the $N!$ permutations of the numbers $(1, \cdots, N)$ with equal probability, $1/N!$. Sufficient conditions are given for the asymptotic normality of $S_N = \sum^N_{i=1}a_{Ni}b_{NR_i}$, where $(a_{N1}, \cdots, a_{NN}), (b_{N1}, \cdots, b_{NN})$ are two sets of real numbers given for every $N$. These sufficient conditions are apparently quite different from those given by Wald and Wolfowitz [9] and extended by various writers [4, 7]. In some situations the conditions given here may be easier to apply than those given previously. The most general conditions available to date appear to be those of Hoeffding [4]. In the examples below, however, is given a case of an $S_N$ which does not satisfy the conditions required by Hoeffding's theorem but which is asymptotically normal by our results.