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?
> > 
> > 
> > 
> 
> 
> 



Recurrence of Poisson binomial distributed random walk
$S\_n$ is a martingale with bounded jumps, and there is a result that it should either converge to a finite limit, or fluctuate, in the sense that $\limsup S\_n=+\infty$, $\liminf S\_n=-\infty$ (this, I guess, should be understood as "recurrence"). See e.g. Theorem (3.1) of Chapter 4 of [Durrett, "Probability, Theory and Examples" (2004)]. If $\sum p\_n = \sum (1-p\_n) = \infty$, this should rule out the first possibility, so the walk will have to be recurrent.


