Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$.

Question

What are some sharp concentration inequalities (i.e tail bounds) for the empirical process $Z_n := \max(X_1,\ldots,X_n)$ ?

Observation

Boucheron and co-workers have results for sub-gaussian and sub-exponential distributions. These results could in principle be used to obtain rather lax tail bounds bounds for $Z_n$. I was wondering one could do better than such an approach.