Assume $(X_n)_{n\geq1} \subseteq \mathbb {Z}$ and $(Y_n)_{n\geq 1} \subseteq \mathbb {Z}$ to be iid, $X_i \sim Y_i$ and such that $S_n=\sum_{i=1}^n(X_i-Y_i)$ is a strongly aperiodic, recurrent random walk. Let $T \subseteq \mathbb{N}$ be a discrete random variable.
With this information can we then say anything about $P(\exists m, \;S_m =T)$?
Thank you.