What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ?$$
My thoughts:
I guess this is supposed to be a counterexample to the strong law of large numbers. I know that the variance of my sequence needs to be unbounded, because otherwise that sequence would fulfill all conditions of the SLLN. I have found both, sequences that fulfill independency and sequences that fulfill the second condition, but i could not imagine any sequence that fulfills both of them. One idea was to use the Cantor, but I haven't found anything useful.