  In this work the $\ell_q$-norms of points chosen uniformly at random in a
centered regular simplex in high dimensions are studied. Berry-Esseen bounds in
the regime $1\leq q < \infty$ are derived and complemented by a non-central
limit theorem together with moderate and large deviations in the case where
$q=\infty$. A comparison with corresponding results for $\ell_p^n$-balls is
carried out as well.
