# Bounding values of a Dirichlet distribution

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 $$0 < \alpha < \beta < 1$$.