Sum of i.i.d random variables equals to infinity

Question

Let $\{X_i\} $ be i.i.d random variables in $\mathbb{R}+$.

When $\sum_{i=1}^{\infty} X_i=\infty$?

Answer 1

There exists $\epsilon > 0$ such that $\mathbb{P}(X_1 \geq \epsilon) > 0$. Then by the 2^nd Borel-Cantelli Lemma,

$$\mathbb{P}(X_n \geq \epsilon \text{ i.o.}) = 1.$$

Now on the event $\{X_n \geq \epsilon \text{ i.o.}\}$,

$$ \sum_{n=1}^{\infty} X_n \geq \sum_{n=1}^{\infty} \epsilon \mathbf{1}_{\{ X_n \geq \epsilon\}} = \infty. $$