## Asymptotic distribution of the Pearson chi-square statistic

I recently learned of a fairly succinct proof for the asymptotic distribution of the Pearson chi-square statistic (from Chapter 9 of Reference 1), which I share below.

First, the set-up: Assume that we have \$latex n\$ independent trials, and each trial ends in one of \$latex J\$ possible outcomes, which we label (without loss of generality) as \$latex 1, 2, dots, J\$. Assume that for each trial, the probability of the outcome being \$latex j\$ is \$latex p_j > 0\$. Let \$latex n_j\$ denote that number of trials that result in outcome \$latex j\$, so that \$latex sum_{j=1}^J n_j = n\$. Pearson’s \$latex chi^2\$-statistic is defined as

\$latex begin{aligned} chi^2 = sum_{text{cells}} dfrac{(text{obs} – text{exp})^2}{text{exp}} = sum_{j=1}^J dfrac{(n_j – np_j)^2}{np_j}. end{aligned}\$

Theorem. As \$latex n rightarrow infty\$, \$latex chi^2 stackrel{d}{rightarrow} chi_{J-1}^2\$, where \$latex stackrel{d}{rightarrow}\$ denotes convergence in distribution.

Before proving the theorem, we prove a lemma that we will…

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