Communications in Mathematical Physics

, Volume 61, Issue 3, pp 293–305 | Cite as

Some limit theorems for random fields

  • Carla C. Neaderhouser


We prove a central limit theorem with remainder and an iterated logarithm law for collections of mixing random variables indexed byZ d ,d≧1. These results are applicable to certain Gibbs random fields.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Limit Theorem 
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  1. 1.
    Deo, C. M.: A note on empirical processes of strong-mixing sequences. Ann. Prob.1, 870–875 (1973)Google Scholar
  2. 2.
    Dobrushin, R. L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theor. Prob. Appl.13, 197–224 (1968)CrossRefGoogle Scholar
  3. 3.
    Dobrushin, R. L., Tirozzi, B.: The central limit theorem and the problem of equivalence of ensembles. Commun. math. Phys.54, 173–192 (1977)CrossRefGoogle Scholar
  4. 4.
    Hegerfeldt, G. C., Nappi, C. R.: Mixing properties in lattice systems. Commun. math. Phys.53, 1–7 (1977)CrossRefGoogle Scholar
  5. 5.
    Helms, L. L.: Semigroups of operators and interacting particles. (unpublished manuscript)Google Scholar
  6. 6.
    Holley R. A., Stroock, D. W.: Applications of the stochastic Ising model to the Gibbs states. Commun. math. Phys.48, 249–265 (1976)CrossRefGoogle Scholar
  7. 7.
    Lebowitz, J. L., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. math. Phys.25, 276–282 (1972)CrossRefGoogle Scholar
  8. 8.
    Loève, M.: Probability theory, 3rd ed., Princeton, New Jersey: Van Nostrand 1963Google Scholar
  9. 9.
    Neaderhouser, C. C.: Limit theorems for multiply-indexed mixing random variables, with applications to Gibbs random fields. Ann. Prob.6, 207–215 (1978)Google Scholar
  10. 10.
    Penrose, O., Lebowitz, J. L.: On the exponential decay of correlation functions. Commun. math. Phys.39, 165–184 (1974)CrossRefGoogle Scholar
  11. 11.
    Philipp, W.: The central limit problem for mixing sequences of random variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete12, 155–171 (1969)CrossRefGoogle Scholar
  12. 12.
    Philipp, W.: The remainder in the central limit theorem for mixing stochastic processes. Ann. Math. Statist.40, 601–609 (1969)Google Scholar
  13. 13.
    Philipp, W., Stout, W. F.: Almost sure invariance principles for partial sums of weakly dependent random variables. Am. Math. Soc. Mem. No. 161. Providence, Rhode Island: American Mathematical Society 1975Google Scholar
  14. 14.
    Stout, W. F.: Almost sure convergence. New York: Academic Press 1974Google Scholar
  15. 15.
    Reznik, M. Kh.: The law of the iterated logarithm for some classes of stationary processes. Theor. Prob. Appl.8, 606–621 (1968)CrossRefGoogle Scholar
  16. 16.
    Volkonskii, V. A., Rozanov, Yu. A.: Some limit theorems for random functions I. Theor. Prob. Appl.4, 178–197 (1959)CrossRefGoogle Scholar
  17. 17.
    Wichura, M.: Some Strassen-type laws of the iterated logarithm for multiparameter stochastic processes with independent increments. Ann. Prob.1, 272–296 (1973)Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Carla C. Neaderhouser
    • 1
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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