A Combinatorial Central Limit Theorem

Let (Yₙ₁, ⋯, Yₙₙ) 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 = ∑ⁿᵢ₌₁ cₙ(i, Yₙᵢ) 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).