Abstract
Some properties of the set of vertices not visited by a random walk on the cube are considered. The asymptotic distribution of the first timeQ this set is empty is derived. The distribution of the number of vertices not visited is found for times nearEQ. Next the first time all unvisited vertices are at least some distanced apart is explored. Finally the expected time taken by the path to come within a distanced of all points is calculated. These results are compared to similar results for random allocations.
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References
Diaconis, P. (1988).Group Theory in Statistics. IMS, Hayward, CA.
Kolchin, V., Sevast'yanov, B., and Christyakov, V. (1978).Random Allocations, V. H. Winston and Sons, New York.
Aldous, D. (1983).On the time taken by random walks on finite groups to visit every state. Z. Wahrsch. Verw. Gebiete62, 361–374.
Matthews, P. (1988). Covering problems for Brownian motion on spheres.Ann. Prob. 16, 189–199.
Matthews, P. (1988). Covering problems for Markov chains.Ann. Prob. 16, 1215–1228.
Kemperman, J. (1961).The Passage Problem for a Stationary Markov Chain. University of Chicago Press, Chicago.
MacWilliams, F., and Sloane, N. (1977).The Theory of Error-Correcting Codes. North-Holland Amsterdam.
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Matthews, P. Some sample path properties of a random walk on the cube. J Theor Probab 2, 129–146 (1989). https://doi.org/10.1007/BF01048275
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DOI: https://doi.org/10.1007/BF01048275