Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Deo, C. M.: A note on empirical processes of strong-mixing sequences. Ann. Prob.1, 870–875 (1973)
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)
Dobrushin, R. L., Tirozzi, B.: The central limit theorem and the problem of equivalence of ensembles. Commun. math. Phys.54, 173–192 (1977)
Hegerfeldt, G. C., Nappi, C. R.: Mixing properties in lattice systems. Commun. math. Phys.53, 1–7 (1977)
Helms, L. L.: Semigroups of operators and interacting particles. (unpublished manuscript)
Holley R. A., Stroock, D. W.: Applications of the stochastic Ising model to the Gibbs states. Commun. math. Phys.48, 249–265 (1976)
Lebowitz, J. L., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. math. Phys.25, 276–282 (1972)
Loève, M.: Probability theory, 3rd ed., Princeton, New Jersey: Van Nostrand 1963
Neaderhouser, C. C.: Limit theorems for multiply-indexed mixing random variables, with applications to Gibbs random fields. Ann. Prob.6, 207–215 (1978)
Penrose, O., Lebowitz, J. L.: On the exponential decay of correlation functions. Commun. math. Phys.39, 165–184 (1974)
Philipp, W.: The central limit problem for mixing sequences of random variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete12, 155–171 (1969)
Philipp, W.: The remainder in the central limit theorem for mixing stochastic processes. Ann. Math. Statist.40, 601–609 (1969)
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 1975
Stout, W. F.: Almost sure convergence. New York: Academic Press 1974
Reznik, M. Kh.: The law of the iterated logarithm for some classes of stationary processes. Theor. Prob. Appl.8, 606–621 (1968)
Volkonskii, V. A., Rozanov, Yu. A.: Some limit theorems for random functions I. Theor. Prob. Appl.4, 178–197 (1959)
Wichura, M.: Some Strassen-type laws of the iterated logarithm for multiparameter stochastic processes with independent increments. Ann. Prob.1, 272–296 (1973)
Author information
Authors and Affiliations
Additional information
Communicated by J. L. Lebowitz
Partially supported by NSF Grant MSC 76-05828 and a grant from the Texas A&M University College of Science
Rights and permissions
About this article
Cite this article
Neaderhouser, C.C. Some limit theorems for random fields. Commun.Math. Phys. 61, 293–305 (1978). https://doi.org/10.1007/BF01940772
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01940772