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} $?


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