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

Question
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What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(X_1,\ldots,X_n)$ ?

Observation
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Boucheron and co-workers have [results for sub-gaussian and sub-exponential distributions][1]. 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.


  [1]: https://arxiv.org/pdf/1207.7209.pdf