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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Distances between two-state Markov processes attainable by Markov joinings

Author: Martin H. Ellis
Journal: Trans. Amer. Math. Soc. 241 (1978), 129-153
MSC: Primary 28A65
MathSciNet review: 0486409
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Abstract: The function which assigns to each pair of two-state Markov processes the set of partition distances between them attainable by a Markov process on their joint atoms is computed. It is found that the infimum of these distances, the ``Markov distance'' between the pair, fails to satisfy the Triangle Inequality, hence fails to be a metric; thus in some cases the $ \overline d $-distance between two two-state Markov processes cannot be attained by a Markov process on their joint atoms.

References [Enhancements On Off] (What's this?)

  • [1] Kenneth R. Berg, Convolution of invariant measures, maximal entropy, Math. Systems Theory 3 (1969), 146–150. MR 0248330
  • [2] Martin H. Ellis, The \overline𝑑-distance between two Markov processes cannot always be attained by a Markov joining, Israel J. Math. 24 (1976), no. 3-4, 269–273. MR 0414820
  • [3] -, Conditions for attaining $ \overline d $ by a Markovian joining (submitted).
  • [4] -, On a conjecture by Kamae (submitted).
  • [5] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 0213508
  • [6] T. Kamae, U. Krengel, and G. L. O’Brien, Stochastic inequalities on partially ordered spaces, Ann. Probability 5 (1977), no. 6, 899–912. MR 0494447
  • [7] John G. Kemeny and J. Laurie Snell, Finite Markov chains, The University Series in Undergraduate Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115196
  • [8] Donald S. Ornstein, An application of ergodic theory to probability theory, Ann. Probability 1 (1973), no. 1, 43–65. MR 0348831
  • [9] Donald S. Ornstein, Ergodic theory, randomness, and dynamical systems, Yale University Press, New Haven, Conn.-London, 1974. James K. Whittemore Lectures in Mathematics given at Yale University; Yale Mathematical Monographs, No. 5. MR 0447525

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Keywords: Stationary stochastic process, Markov process, joint process, partition distance, $ \overline d $-distance, Markov joining, attaining $ \overline d $, Markov joining attaining $ \overline d $, distances attainable by Markov joinings
Article copyright: © Copyright 1978 American Mathematical Society