Central Limit Theorems for Multinomial Sums

Let (N₁, ⋯, Nₖ) be a multinomial vector with n = ∑ Nᵢ and with parameter (p₁, ⋯, pₖ), ∑ pᵢ = 1. Let f₁, ⋯, fₖ be real-valued functions defined on the integers \0, 1, ⋯, n\, and let Sₖ = ∑ᵏᵢ₌₁ fᵢ(Nᵢ). Suppose k → ∞, and as k → ∞, that n → ∞ and max1 i kpᵢ → 0. Conditions on the ᵢ\ are given which guarantee that Sₖ, suitably centered and scaled, has a normal limit in law. An application shows that if min1 i k (npᵢ) is bounded away from zero and the ᵢ\ are polynomials of bounded degree as k → ∞, that Sₖ is asymptotically normal provided only that a "uniformly asymptotically negligible" (uan) condition on the ᵢ\ holds. For testing the specified simple hypothesis pᵢ = pᵢ⁰ for all 1 i k, Pearson's "chi-square" statistic and the likelihood ratio statistic can be written in the form of Sₖ. It is shown that these two statistics are asymptotically normal as k → ∞ provided they satisfy simple conditions which are equivalent to their respective uan conditions.