Let $X_n$ be the outcome of a Bernoulli trial where the probability of getting 1 is $p_n$ and the probability of getting 0 is $1-p_n$, and let $S_n = \sum_{i=1}^n \left(X_i - \textrm{E} X_i \right)$. In general, $p_i \neq p_j$, so $S_n$ is Poisson binomial distributed, but with the mean subtracted. Since $S_n$ has finite variance and expectation value 0, I would assume that $S_n$ is recurrent and that this should be a fairly well-known result. However, I have been browsing the literature and asking around, and despite the problem being apparently rather simple, I have not yet been able to find a proof.

QUESTION: Does anyone have a proof that $S_n$ is recurrent, alternatively, references to relevant literature where I could find proof(s) and discussion(s) of this problem?