Let $\{D_i\}_{i=0,1,2,\dots }$ be independent $\exp(1)$ random variables. We use the collection $\{D_i\}$ to define a random walk on $\mathbb Z$ by $S_0 = 0$ and $S_n = \sum_1^n X_i$ with $X_i \in \{-1,0,1\}$ given by $$X_i = \mathbf 1 \{ D_{i} > 2 D_{i+1} \} - \mathbf 1 \{ D_{i+1} > 2 D_{i } \}.$$ I am interested in showing $S_n$ visits 0 infinitely often a.s. The pairwise dependence between $X_i$ and $X_{i+1}$ is making this (frustatingly) difficult.
The assumption that $D_i \sim \exp(1)$ is just for convenience and should not be important.