Convergence of a series of random variables

Question

As I usually say in my posts, I'm definitely not looking for an answer, just a hint!

Suppose I know that $(X_n)$ is a sequence of i.i.d. mean $0$ random variables, all bounded by some constant $k$, where $\mathbb P[\forall n,\sum_{m\leq n}X_m\textrm{ is bounded}]>0$. I want to see if $\sum_{n\geq0}X_n$ converges.

I definitely missed some material in class, because I don't even have a vague idea of what I should be looking for. In a way, I feel like I need something close to the Borel-Cantelli lemmas, but the actual ones obviously don't work.

Again, no answers, just a hint.

Answer 1

I think I have a solution.

Consider the family of $\sigma$-algebras $(\sigma(X_n))$. Generate a tail $\sigma$-algebra $\mathcal T$ on this family. Now observe that Kolmogorov's 0-1 law guarantees us that $\mathbb P(\sum_{n\geq0}X_n\textrm{ exists})=0$ or $1$ since $\{\sum_{n\geq0}X_n\textrm{ exists}\}\in\mathcal T$.

However, as $0<\mathbb P(\textrm{all partial sums }\sum_{m\leq n}X_m\textrm{ are bounded})\leq\mathbb P(\sum_{n\geq0}X_n\textrm{ exists})$ (which holds because all partial sums being bounded is necessary for the series to exist) we must conclude that $\mathbb P(\sum_{n\geq0}X_n\textrm{ exists})=1$ or that $\sum_{n\geq0}X_n$ converges almost surely.

Please inform me if I'm wrong.