If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know $$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$ where $\bf{X}=(X_1,\ldots,X_k)^T$ and $p=(p_1,\ldots,p_k)^T$.

What if now $k$ is an increasing function of $n$ (for example, $k=n^{1/2}$), then what is the corresponding limiting distribution?


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