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Consider $k$ random variables $X_1, X_2, \ldots, X_k$ such that $(X_1, X_2, \ldots, X_k)$ follow a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution. For a large enough $k$, I am trying to bound/find out (with high probability) what fraction of these $k$ random variables take values between $\alpha$ and $\beta$, for $0 < \alpha < \beta < 1$.

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