# Random walk probability question.

Consider two simple random walk processes, i.e. we have $$\{S_{n}' = X_{1}' + \dots + X_{n}'\}$$ and $$\{ S_{n}' = X_{1} + \dots + X_{n}\}$$ on $$\mathbb{Z}^d$$ ($$\mathbb{P}(X_{i} = e_k) = \mathbb{P}(X_{i} = -e_k) = \mathbb{P}(X'_{i} = e_k) = \mathbb{P}(X'_{i} = -e_k) = \frac{1}{2d}$$ and $$X_{i}$$ , $$X'_{i}$$ are independent).

As we know Polya's theorem state that random walk is returnable iff $$d=1,2$$.

But for which $$d$$ we have $$\mathbb{E} \sum \operatorname{1}(S_n = S'_m) < \infty$$?

I was thought about : $$\sum_{n,m} \mathbb{P}(S_n = S'_m, S_n = \bar{x}, S'_m = \bar{x})$$. But there is a problem with counting $$\bar{x}$$ in integer space. Maybe it's better to focus on $$Q_{n,m}=S_{n} - S'_{m}$$?