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Let (Y₍₁, , Y₍₍) be a random vector which takes on the n! permutations of (1, , n) with equal probabilities. Let cₙ (i, j), i, j = 1, , n, be n² real numbers. Sufficient conditions for the asymptotic normality of Sₙ = ⁿ₈=₁ cₙ (i, Y₍₈) are given (Theorem 3). For the special case cₙ (i, j) = aₙ (i) bₙ (j) a stronger version of a theorem of Wald, Wolfowitz and Noether is obtained (Theorem 4). A condition of Noether is simplified (Theorem 1).
Wassily Hoeffding (Sat,) studied this question.