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.
Citation
Meyer Dwass. "On the Asymptotic Normality of Certain Rank Order Statistics." Ann. Math. Statist. 24 (2) 303 - 306, June, 1953. https://doi.org/10.1214/aoms/1177729038
Information