Abstract
Let S be a non-empty set. S may be either finite or countably infinite. Elements of S are referred to as sites. We wish to consider random distributions of ±1 values on S. The configuration space is the product space Ω = {−1,1}S equipped with the Borel sigma-field ⌁ for the product topology on Ω when {−1,1} has the discrete topology, Equivalently, ⌁ is the sigma-field generated by the finite-dimensional events. Let ℒ denote the collection of finite subsets of S.
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References
Burton, R. and E. Waymire, Scaling limits for associated random measures. Ann. of Prob., in press, (1985).
Burton, R. and E. Waymire, The central limit problem for infinitely divisible random measures, This edition, (1985).
Esary, J., Proschan F., and D. Walkup, Association of random variables with applications. Ann. Math. Stat., 58 (1967), 1466–1474.
Fortuin, C., Kasteleyn, P., and J. Ginibre, Correlation inequalities on some partially ordered sets. Comm. Mth. Phys., 22 (1971), 89–103.
Glaffig, C. and E. Waymire, Infinite divisibility of the Bethe lattice Ising model. Preprint (1985).
Griffiths, R.B., Correlations in Ising ferromagnets. J. Math. Phys., 8 (1967), 478–489.
Griffiths, R.B., Hurst, C.A., and S. Sherman, Concavity of magnetization of an Ising ferro-magnet in a positive external field. J. Math. Phys., 11 (1970), 790–795.
Grimmet, G.R., A theorem about random fields. Bull. London Math. Soc., 5 (1973), 81–84.
Harris, T.E., A correlation inequality for Markov processes in partially ordered spaces. Ann. of Prob., 5 (1977), 451–454.
Holley, R., Remarks on the FKG inequalities. Comm. Math. Phys., 36 (1974), 227–231.
Ising, E., Beitrag sur theorie des ferromagnetismus. Zeit, fur Physik, 31 (1925), 253–258.
Kelly, D.G. and S. Sherman, General Griffiths inequalities on correlations in Ising ferro-magnets. J. Math. Phys., 9 (1968), 466–484.
Liggett, T.M., Interacting Particle Systems. Springer-Verlag, New York, 1985.
Newman, C.M., Asymptotic independence and limit theorems for positively and negatively dependent random variables. Inequalities in Statistics and Probability, IMS Lecture Notes, 1984.
Parthasarathy, K.R., Probability measures on metric spaces. Academic Press, New York, 1967.
Preston, C., Generalized Gibbs states and Markov random fields. Adv. in Appld. Prob., 5 (1973), 242–261.
Preston, C., Gibbs states on countable sets. Cambridge Univ. Press, 1974.
Preston, C., Random Fields. Springer-Verlag, No.534, New York, 1976.
Rudin, W., Fourier analysis on groups. Wiley, New York, 1962.
Sherman, S., Markov random fields and Gibbs random fields. Israel J. Math., 14 (1973), 92–103.
Spitzer, F., Random fields and interacting particle systems. M.A.A. Summer Seminar Notes, 1971.
Sptizer, F., Markov random fields on an infinite tree. Ann. of Prob., 3 (1975), 387–398.
Sullivan, W.G., Potentials for almost Markovian random fields. Comm. Math. Phys., 133 (1973), 61–74.
Taqqu, M., Self-similar processes and long-range dependence: a bibliographical survey. This edition, 1985.
Waymire, E., Infinitely divisible Gibbs states. Rcky. Mtn. J. Math., 14(3) (1984), 665–678.
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Waymire, E. (1986). Infinitely Divisible Distributions; Gibbs States and Correlations. In: Eberlein, E., Taqqu, M.S. (eds) Dependence in Probability and Statistics. Progress in Probability and Statistics, vol 11. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4615-8162-8_16
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DOI: https://doi.org/10.1007/978-1-4615-8162-8_16
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