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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) empirical statistic defined by $Z_n := \max(X_1,\ldots,X_n)$ ?

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.

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For the tail of $Z_n$, we have the very simple exact formula $P(Z_n>t)=1-F(t)^n$ for all real $t$, where $F$ is the cdf of $X_1$. Do you want a bound on it? Of what kind?

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  • $\begingroup$ The formula you're proposing is valid for the uniform distribution on $[0,1]$. My problem is more general. The distribution is only required to supported on $[0, 1]$, not necessarily uniform thereupon. Agreed ? Concerning the remark about my use of "empirical processé", it's in fact not a very interesting process. Fixed. $\endgroup$ – dohmatob Dec 20 '18 at 20:19
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    $\begingroup$ Oops, I missed it that the distribution on $[0,1]$ may arbitrary. This does not change the answer much, though. $\endgroup$ – Iosif Pinelis Dec 20 '18 at 21:18

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