Extreme value distributions of uncountably infinite set of random variables

Question

Let us suppose that we have an uncountably infinite set $A=\{x_1,x_2, \cdots\}$ of i.i.d. random variables $x_i$, say with gamma distribution. Are minimum and maximum extreme value distributions exist for this set? What I know is the extreme value distributions of the countably infinite set. I am not sure about the uncountably infinite set. Any ideas or pointers would be appreciated.

Answer 1

The probability that $\min (X_1, \dots, X_n) > \epsilon$ for any $\epsilon > 0$ is equal to $$e_n = \left(\int_\epsilon^\infty f(x) dx\right)^n,$$ where $f$ is the p.d.f. of the Gamma distribution, since each $X_i$ must be greater than $\epsilon$ for this to be true. Since $0 < \int_\epsilon^\infty f(x) dx < 1$, we have $\lim_{n \rightarrow \infty} e_n = 0$. So there are guaranteed to be elements of $\{x_1, \dots\}$ lower than any $\epsilon > 0$. Similarly for $\max(X_1, \dots, X_n)$, if we consider $$m_n = \mathbb{P}(\max(X_1, \dots, X_n) < M) = \left(\int_0^M f(x) dx\right)^n$$ for any $M > 0$ we have $\lim_{n \rightarrow \infty} m_n = 0$, so there are guaranteed to be elements of $\{x_1, \dots\}$ that exceed any $M > 0$.

There will be no minimum or maximum element of $\{x_1, x_2, \dots\}$, but the set will have infimum $0$ and supremum $\infty$.