= Multinomial central limit theorem
{title2=$\sqrt n(\widehat p-p)\Rightarrow N(0,\operatorname{diag}p-pp^T)$}
The vector of category counts is the sum of independent categorical indicator vectors. Their means are $p$ and covariance is $\operatorname{diag}p-pp^T$, so the <multivariate central limit theorem> gives the displayed limit. At uniform probabilities on $k$ categories, Pearson-standardized deviations have covariance equal to the orthogonal projection onto the $(k-1)$-dimensional subspace of zero-sum vectors. Their squared norm has an asymptotic <chi-squared distribution> with $k-1$ degrees of freedom.
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