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Markov approximation of homogeneous lattice random fields
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  • Published: 12 September 2004

Markov approximation of homogeneous lattice random fields

  • B.M. Gurevich1 &
  • A.A. Tempelman2 

Probability Theory and Related Fields volume 131, pages 519–527 (2005)Cite this article

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  • 7 Citations

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Abstract.

We refine some well-known approximation theorems in the theory of homogeneous lattice random fields. In particular, we prove that every translation invariant Borel probability measure μ on the space X of finite-alphabet configurations on ℤd, d≥1, can be weakly approximated by Markov measures μ n with supp(μ n )=X and with the entropies h(μ n )→h(μ). The proof is based on some facts of Thermodynamic Formalism; we also present an elementary constructive proof of a weaker version of this theorem.

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Acknowledgments.

(1). The authors are deeply indebted to the remarkable doctors, Prof. A. I. Vorobyov and E. A. Gilyazitdinova. Without their skill and their concern for the first author this paper would have not been written. (2). We are grateful to B. Weiss for calling our attention to the ‘‘synchronizing prefix’’ method used in Section 3. (3). The remarks by the referees resulted in essential improvement of the text of the paper. (4). The first author is supported by the RFBR grant number 02-01-00444.

Author information

Authors and Affiliations

  1. Leninskie Gory, Department of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia

    B.M. Gurevich

  2. Department of Statistics, Penn State University, 325 Thomas Bldg., University Park, PA, 16802, USA

    A.A. Tempelman

Authors
  1. B.M. Gurevich
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  2. A.A. Tempelman
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Corresponding author

Correspondence to A.A. Tempelman.

Additional information

Mathematics Subject Classifications (2000): Primary 28D20, 37C85, 60G60; secondary 82B20

Dedicated to Professor A. I. Vorobyov, member of the Russian Academy of Sciences and Director of the Hematology Research Center of the Russian Academy of Medical Sciences, on the occasion of his 75th birthday

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Cite this article

Gurevich, B., Tempelman, A. Markov approximation of homogeneous lattice random fields. Probab. Theory Relat. Fields 131, 519–527 (2005). https://doi.org/10.1007/s00440-004-0383-6

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  • Received: 20 November 2003

  • Revised: 10 May 2004

  • Published: 12 September 2004

  • Issue Date: April 2005

  • DOI: https://doi.org/10.1007/s00440-004-0383-6

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Keywords

  • Homogeneous random fields
  • Markov fields
  • Equilibrium measures
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