The Annals of Probability

Wiener's test for random walks with mean zero and finite variance

K\^{o}hei Uchiyama

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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 $N \geq 4$.

Article information

Ann. Probab., Volume 26, Number 1 (1998), 368-376.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60J15 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 31C20: Discrete potential theory and numerical methods

Multidimensional random walk Green's function Laplace discrete operator Wiener's test Markov chain transient set


Uchiyama, K\^{o}hei. Wiener's test for random walks with mean zero and finite variance. Ann. Probab. 26 (1998), no. 1, 368--376. doi:10.1214/aop/1022855424.

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