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?
Best Answer
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.