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 $Y:=\underset{i\geq 1}{\max}~X_i$?
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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.

$\begingroup$ So this proof suggests that the distribution of an infinite series i.i.d. random variables is always the maximum possible value of a single random variable with probability 1? $\endgroup$ – kim kevin Feb 7 '17 at 1:29

$\begingroup$ Yeah this is not particular to the Gumbel distribution. $\endgroup$ – Bjørn KjosHanssen Feb 7 '17 at 3:01