# Sharp tail bounds for the maximum of an iid sample of a random variable supported on $[0, 1]$

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.

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?
• 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. Dec 20 '18 at 20:19
• Oops, I missed it that the distribution on $[0,1]$ may arbitrary. This does not change the answer much, though. Dec 20 '18 at 21:18