Infinitely Divisible Distributions; Gibbs States and Correlations

  • Ed Waymire
Part of the Progress in Probability and Statistics book series (PRPR, volume 11)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Burton, R. and E. Waymire, Scaling limits for associated random measures. Ann. of Prob., in press, (1985).Google Scholar
  2. [2]
    Burton, R. and E. Waymire, The central limit problem for infinitely divisible random measures, This edition, (1985).Google Scholar
  3. [3]
    Esary, J., Proschan F., and D. Walkup, Association of random variables with applications. Ann. Math. Stat., 58 (1967), 1466–1474.CrossRefMathSciNetGoogle Scholar
  4. [4]
    Fortuin, C., Kasteleyn, P., and J. Ginibre, Correlation inequalities on some partially ordered sets. Comm. Mth. Phys., 22 (1971), 89–103.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    Glaffig, C. and E. Waymire, Infinite divisibility of the Bethe lattice Ising model. Preprint (1985).Google Scholar
  6. [6]
    Griffiths, R.B., Correlations in Ising ferromagnets. J. Math. Phys., 8 (1967), 478–489.CrossRefGoogle Scholar
  7. [7]
    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.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Grimmet, G.R., A theorem about random fields. Bull. London Math. Soc., 5 (1973), 81–84.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Harris, T.E., A correlation inequality for Markov processes in partially ordered spaces. Ann. of Prob., 5 (1977), 451–454.CrossRefzbMATHGoogle Scholar
  10. [10]
    Holley, R., Remarks on the FKG inequalities. Comm. Math. Phys., 36 (1974), 227–231.CrossRefMathSciNetGoogle Scholar
  11. [11]
    Ising, E., Beitrag sur theorie des ferromagnetismus. Zeit, fur Physik, 31 (1925), 253–258.CrossRefGoogle Scholar
  12. [12]
    Kelly, D.G. and S. Sherman, General Griffiths inequalities on correlations in Ising ferro-magnets. J. Math. Phys., 9 (1968), 466–484.CrossRefGoogle Scholar
  13. [13]
    Liggett, T.M., Interacting Particle Systems. Springer-Verlag, New York, 1985.CrossRefzbMATHGoogle Scholar
  14. [14]
    Newman, C.M., Asymptotic independence and limit theorems for positively and negatively dependent random variables. Inequalities in Statistics and Probability, IMS Lecture Notes, 1984.Google Scholar
  15. [15]
    Parthasarathy, K.R., Probability measures on metric spaces. Academic Press, New York, 1967.zbMATHGoogle Scholar
  16. [16]
    Preston, C., Generalized Gibbs states and Markov random fields. Adv. in Appld. Prob., 5 (1973), 242–261.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    Preston, C., Gibbs states on countable sets. Cambridge Univ. Press, 1974.CrossRefzbMATHGoogle Scholar
  18. [18]
    Preston, C., Random Fields. Springer-Verlag, No.534, New York, 1976.Google Scholar
  19. [19]
    Rudin, W., Fourier analysis on groups. Wiley, New York, 1962.zbMATHGoogle Scholar
  20. Sherman, S., Markov random fields and Gibbs random fields. Israel J. Math., 14 (1973), 92–103.Google Scholar
  21. [21]
    Spitzer, F., Random fields and interacting particle systems. M.A.A. Summer Seminar Notes, 1971.Google Scholar
  22. [22]
    Sptizer, F., Markov random fields on an infinite tree. Ann. of Prob., 3 (1975), 387–398.CrossRefGoogle Scholar
  23. [23]
    Sullivan, W.G., Potentials for almost Markovian random fields. Comm. Math. Phys., 133 (1973), 61–74.CrossRefGoogle Scholar
  24. [24).
    Taqqu, M., Self-similar processes and long-range dependence: a bibliographical survey. This edition, 1985.Google Scholar
  25. [25]
    Waymire, E., Infinitely divisible Gibbs states. Rcky. Mtn. J. Math., 14(3) (1984), 665–678.Google Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Ed Waymire
    • 1
  1. 1.Department of MathematicsOregon State UniversityCorvallisUSA

Personalised recommendations