Abstract
Before ItÔ and Mckean characterized the recurrent and transient sets for the simple random walk in 3 dimensions, it was thought that a condition of the form
might be necessary and sufficient for B to be recurrent. Their characterization has been extended to hold for an arbitrary 3-dimensional aperiodic random walk with zero mean and finite second moments; in this paper it is used to show that for such a random walk no condition of type (A) can be necessary and sufficient for B to be recurrent, and to find the best possible conditions of type (A) which are necessary or sufficient for B to be recurrent.
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References
Blackwell, D.: On transient Markov processes with a countable number of states and stationary transition probabilities. Ann. math. Statistics 26, 654–658 (1955).
Breiman, L.: Transient atomic Markov chains with a denumerable number of states. Ann. math. Statistics 29, 212–218 (1958).
ItÔ, K., and H. P. Mckean, Jr.: Potentials and the random walk. Illinois J. Math. 4, 119–132 (1960).
Spitzer, F.: Principles of random walk. New York: Van Nostrand 1964.
Doney, R. A.: Thesis. Durham University, 1964.
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I am grateful to Professor G. E. H. Reuter for much helpful advice, and to the Department of Scientific and Industrial Research for financial support.
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Doney, R.A. Recurrent and transient sets for 3-dimensional random walks. Z. Wahrscheinlichkeitstheorie verw Gebiete 4, 253–259 (1965). https://doi.org/10.1007/BF00533756
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DOI: https://doi.org/10.1007/BF00533756