Some Extensions of the Wald-Wolfowitz-Noether Theorem

Abstract

Let $(R_{u 1}, \cdots, R_{{u N}_u})$ be a random vector which takes on the $N_u!$ permutations of $(1, \cdots, N_u)$ with equal probabilities. Let $\{b_{u i}, 1 \leqq i \leqq N_u, v \geqq 1\}$ and $\{a_{u i}, 1 \leqq i \leqq N_u, v \geqq 1\}$ be double sequences of real numbers. Put \begin{equation*}\tag{1.1}S_u = \sum^{N_u}_{i = 1} b_{u i}a_{u R_{u i}}.\end{equation*} We shall prove that the sufficient and necessary condition for asymptotic $(N_u \rightarrow \infty)$ normality of $S_u$ is of Lindeberg type. This result generalizes previous results by Wald-Wolfowitz [1], Noether [3], Hoeffding [4], Dwass [6], [7] and Motoo [8]. In respect to Motoo [8] we show, in fact, that his condition, applied to our case, is not only sufficient but also necessary. Cases encountered in rank-test theory are studied in more detail in Section 6 by means of the theory of martingales. The method of this paper consists in proving asymptotic equivalency in the mean of (1.1) to a sum of infinitesimal independent components.