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The Gumbel distribution term in Wikipedia says:

Gumbel has shown that the maximum value (or last order statistic) in a sample of a random variable following an exponential distribution approaches the Gumbel distribution closer with increasing sample size.

And this is confusing because the support of Gumbel distribution is real line but that of exponential distribution is positive reals. How could the convergence (in distribution) even appear? The reference link seems to be invalid.

I tried to prove it as well.

WLOG we assume a sample of iid standard exponential random variable of size n. The cdf of the last order stats is given by: $(1-\exp(-x))^n$ on positive reals. The cdf of a standard Gambel distribution is $\exp(-(\exp(-x)))$ on real line. How can we prove the convergence?

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    $\begingroup$ Is it any weirder than the sample mean approaching a Normal distribution? $\endgroup$ Commented Jan 29, 2020 at 22:53
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    $\begingroup$ I think it should have something to do with$$\lim_{n\to \infty} (1-{e^{-x}\over n})^n=e^{-e^{-x}}$$ $\endgroup$ Commented Jan 29, 2020 at 22:59
  • $\begingroup$ Good point from Mostafa Ayaz! Yet the general formula of cdf of exp random variable is $1-exp(-\lambda x)$ with nothing multiplied on exponent part. That may implies that the original statement is not about iid exp variable at all. They may consists a sequence of exp variable with certain pattern of parameters I guess. $\endgroup$ Commented Jan 29, 2020 at 23:13
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    $\begingroup$ Yes. Consider $\dfrac{ X_{(n)} - a_n}{ b_n }$ where $a_n$ and $b_n$ are shifting and scaling constants that depends on $n$. It's easy to figure out that only $a_n \propto \log n$ will allow convergence to a distribution that is neither point mass nor trivial, and the result will be a Gumbel which parameter depends on the limit of $b_n$. $\endgroup$ Commented Jan 29, 2020 at 23:32
  • $\begingroup$ You can get the proper font for $\exp$ and $\log$ using \exp and \log. For operators that don't have a command of their own, you can use \operatorname{name}. $\endgroup$ Commented Jan 30, 2020 at 2:07

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Thanks to StubbornAtom's comment. It turns out to be duplicated with Convergence in distribution of maximum of exponentially distributed random variables

Instead of the convergence of $\max_{1 \leq k \leq n} X_{k}$, it should be that of $\max_{1 \leq k \leq n} X_{k} - \log(n)$, which may need to be mentioned in the Wikipedia term.

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