0
$\begingroup$

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

$\endgroup$

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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
    Commented Feb 7, 2017 at 1:29
  • $\begingroup$ Yeah this is not particular to the Gumbel distribution. $\endgroup$ Commented Feb 7, 2017 at 3:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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