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Finally I would like to express my thanks to Dr. T.P. Speed for many helpful suggestions.
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Newbould, M. A classification of a random walk defined on a finite Markov chain. Z. Wahrscheinlichkeitstheorie verw Gebiete 26, 95–104 (1973). https://doi.org/10.1007/BF00533479
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DOI: https://doi.org/10.1007/BF00533479