Some Extensions of the Wald-Wolfowitz-Noether Theorem

Abstract

Let $(R_{\nu 1}, \cdots, R_{{\nu N}_\nu})$ be a random vector which takes on the $N_\nu!$ permutations of $(1, \cdots, N_\nu)$ with equal probabilities. Let $\{b_{\nu i}, 1 \leqq i \leqq N_\nu, v \geqq 1\}$ and $\{a_{\nu i}, 1 \leqq i \leqq N_\nu, v \geqq 1\}$ be double sequences of real numbers. Put \begin{equation*}\tag{1.1}S_\nu = \sum^{N_\nu}_{i = 1} b_{\nu i}a_{\nu R_{\nu i}}.\end{equation*} We shall prove that the sufficient and necessary condition for asymptotic $(N_\nu \rightarrow \infty)$ normality of $S_\nu$ 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.