I have an infinite series of independent identically distributed random variables $\{X\_i\}\_{i=1}^\infty$ which follows extreme value type I distribution which can be found [here] (<https://en.wikipedia.org/wiki/Gumbel_distribution>), then I was wondering what is the distribution of $Y:=\underset{i\geq 1}{\max}~X\_i$?



The distribution of the maximum of a series of extreme value type I random variable
For any $n$,
$$\Pr(Y\le n)=\Pr(X\_i\le n\,(\forall i))=\prod\_i\Pr(X\_i\le n)=\lim\_{i\to\infty}\Pr(X\_1\le n)^i=0.$$
So $Y=\infty$ with probability 1.


