Suppose you have a sequence of bernoulli variables, $X_n$ with $\frac{1}{n} = \mathbb{P}(X_n = 1) = 1 - \mathbb{P}(X_n = 0)$ and let $S_n = \sum_{k=1}^{n} X_k X_{k+1} $. The goal is that $S = \lim_{n \rightarrow \infty} S_n$ is Poisson distributed with parameter 1. It's just dependent enough so that we can't apply the central limit theorem for triangular arrays. A friend and I were asked to prove this and we've been racking our heads against the wall with this.
Initially, I thought it might be false, but I computed the variance and some magical cancellation happens so that it gives you the right thing.
I have no idea how to proceed and most arguments I try end up becoming too complicated. Is there something obvious that I'm missing? All help is appreciated!