I encountered a problem recently, which was stated as follows: "suppose $x_i$, $x_2$, ... are a sequence of independent random variables, and let $S_n = x_1 + ... + x_n$. What is the probability that $S_n = 0$ infinitely often?"
I suspect that the Borel-Cantelli Lemma (or its converse) would be used here; however, to me it seems the question is ill-defined, as the solution depends on the sample space of the $x_i$'s. My question, then, is this: is there a solution to this problem as stated, and if so, how can one go about determining (or bounding) $P(S_n = 0)$ for each $n$ to use Borel-Cantelli?
Since $\limsup_{n\to\infty}\{S_n=0\}$ is a tail event, it has probability zero or one. Either case is possible. For example, let $X_n=0$, then clearly $S_n=0$ for all $n$ so $$\mathbb P\left(\limsup_{n\to\infty}\{S_n=0\}\right) = 1.$$ For the other case, let $X_n=1$. Then clearly $\mathbb P(\{S_n=0\})=0$ so $$\mathbb P\left(\limsup_{n\to\infty}\{S_n=0\}\right) = 0.$$