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


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


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?

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.