Central Limit Theorems for Multinomial Sums

The Annals of Statistics · 1975 · Volume 3 · Issue 1

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.

Keywords

62E20, 62F05, Central limit theorem for multinomial sums, hypothesis testing with many parameters, likelihood ratio statistic for the multinomial distribution, multinomial distribution, Pearson's chi-square statistics

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