Some Extensions of the Wald-Wolfowitz-Noether Theorem

Let (Rν 1, ⋯, R_ν N_ν) be a random vector which takes on the N_ν! permutations of (1, ⋯, N_ν) with equal probabilities. Let ν i, 1 i N_ν, v 1\ and ν i, 1 i N_ν, v 1\ be double sequences of real numbers. Put {equation*}{1.1}S_ν = ∑N_νi = 1 bν ia_{ν Rν i}.{equation*} We shall prove that the sufficient and necessary condition for asymptotic (N_ν → ∞) normality of S_ν 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.