I cam across this problem and cannot figure out how to approach or solve it:
Given the random walk $(S_n^z)_{n \in \mathbb{N}}$ starting at $z \in \mathbb{Z}^3$ with uniform distribution on its neighbors. Let $\mathbb{P}_z$ be the probability if we start from a point $z \in \mathbb{Z}^3$. I already know that the random walk is transient for every starting point $z$. But is it also true, that $$\limsup_{|z| \rightarrow \infty}\mathbb{P}_z(\cap_{t \in \mathbb{N}}\{|S_t| > 1\})=1,$$ i.e. that there exist starting points arbitrary far away such that we never enter the units sphere?
Thanks a lot in advance!