Some Extensions of the Wald-Wolfowitz-Noether Theorem

Let (R ₁, , R ₍_) be a random vector which takes on the N_! permutations of (1, , N_) with equal probabilities. Let \b ₈, 1 i N_, v 1\ and \a ₈, 1 i N_, v 1\ be double sequences of real numbers. Put equation*1. 1S_ = ^N_₈ = ₁ b ₈a ₑ_ ₈. equation* We shall prove that the sufficient and necessary condition for asymptotic (N_) normality of S_ 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.