Abstract
It is shown that an infinite subset of $Z^N$ is either recurrent for each aperiodic $N$-dimensional random walk with mean zero and finite variance, or transient for each of such random walks. This is an exact extension of the result by Spitzer in three dimensions to that in the dimensions $N \geq 4$.
Citation
K\^{o}hei Uchiyama. "Wiener's test for random walks with mean zero and finite variance." Ann. Probab. 26 (1) 368 - 376, January 1998. https://doi.org/10.1214/aop/1022855424
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