Distribution of infimum of countably many random variables.

Question

Motivated by my undergraduate class in probability, where we are learning about order statistics: say, you have a countable collection of i.i.d. random variables $X = \{X_1,X_2...\}$. Can we find the distribution of $Y$, where $Y = \inf X$? Take, for an example, $X \sim$ uniform(1,2), if needed.

Answer 1

The random variable $Y = \inf_i X_i$ is almost surely a constant $c$, equal to the essential infimum of the common distribution of the $X_i$. More concretely, if the $X_i$ have cdf $F$, then $Y = c = \inf\{ x : F(x) > 0\}$.

To see why, first notice that $P(Y < c) = 0$. Next, for arbitrary $n,k$, compute the probability (in terms of $F$) that at least one of $X_1, \dots, X_n$ is less than $c+1/k$. This provides a lower bound for $P(Y < c+1/k)$. Now let $n \to \infty$ to find $P(Y < c+1/k)$, and let $k \to \infty$ using countable additivity.

It is possible that $c = -\infty$, in which case this proof needs some minor adjustments which I leave as an exercise.