Abstract
We prove that the one-dimensional random fields with finite first moment are isomorphic to Bernoulli schemes.
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References
Doob, J. L.: Stochastic Processes. New York 1953.
Dobrushin, R. L.: Funct. Anal. Appl.2, 302 (1968) (Section 3, Theorem 2) see also Ruelle, D. Ref. [4] below.
This result, in the particular caser=1, was proven by F. Spitzer: Am. Math. Monthly, February, 142 (1971) and, independently, by M. B. Averintzer (see Reference at p. 143 of Spitzer's paper). The general result is clearly implicitely proved in Griffiths, R. B., Ruelle, D.: Commun. math. Phys.23, 169 (1971) (Section 3, p. 173). An explicit proof can be found in the paper by Del Grosso, G., Tesei, A.: The local central limit theorem for Gibbs' procesus, preprint, Istituto di Matematico, Roma.
Ruelle, D.: Commun. math. Phys.9, 267 (1968).
Friedman, N. A., Ornstein, D. S.: Adv. Math.5, 365 (1970). See also P. Shields: The theory of Bernoulli shifts, Preprint, Univ. of California, Math. dept., Berkeley.
This problem was explicitly raised by J. P. Conze in his talk at the Seminaire Bourbaki, no. 240; «Le théorème d'isomorphisme d'Ornstein et la classification des systèmes dinamiques en theorie ergodique».
Gallavotti, G., Lin, T. F.: Arch. Rat. Mech. Anal.37, 181 (1970).
Dyson, F.: Commun. math. Phys.12, 91 (1969).
Ruelle, D.: Ann. Phys.69, 364 (1972).
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Partially supported by the «Consiglio Nazionale delle Ricerche, GNFM».
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Gallavotti, G. Ising model and Bernoulli schemes in one dimension. Commun.Math. Phys. 32, 183–190 (1973). https://doi.org/10.1007/BF01645655
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DOI: https://doi.org/10.1007/BF01645655