Central Limit Theorems for Multinomial Sums

Abstract

Let $(N_1, \cdots, N_k)$ be a multinomial vector with $n = \sum N_i$ and with parameter $(p_1, \cdots, p_k), \sum p_i = 1$. Let $f_1, \cdots, f_k$ be real-valued functions defined on the integers $\{0, 1, \cdots, n\}$, and let $S_k = \sum^k_{i=1} f_i(N_i)$. Suppose $k \rightarrow \infty$, and as $k \rightarrow \infty$, that $n \rightarrow \infty$ and $\max_{1\leqq i \leqq k}p_i \rightarrow 0$. Conditions on the $\{f_i\}$ are given which guarantee that $S_k$, suitably centered and scaled, has a normal limit in law. An application shows that if $\min_{1\leqq i \leqq k} (np_i)$ is bounded away from zero and the $\{f_i\}$ are polynomials of bounded degree as $k \rightarrow \infty$, that $S_k$ is asymptotically normal provided only that a "uniformly asymptotically negligible" (uan) condition on the $\{f_i\}$ holds. For testing the specified simple hypothesis $p_i = p_i^0$ for all $1 \leqq i \leqq k$, Pearson's "chi-square" statistic and the likelihood ratio statistic can be written in the form of $S_k$. It is shown that these two statistics are asymptotically normal as $k \rightarrow \infty$ provided they satisfy simple conditions which are equivalent to their respective uan conditions.