Abstract
The partial-sum processes, indexed by sets, of a stationary nonuniform φ-mixing random field on the d-dimensional integer lattice are considered. A moment inequality is given from which the convergence of the finite-dimensional distributions to a Brownian motion on the Borel subsets of [0, 1]d is obtained. A Uniform CLT is proved for classes of sets with a metric entropy restriction and applied to certain Gibbs fields. This extends some results of Chen(5) for rectangles. In this case and when the variables are bounded a simpler proof of the uniform CLT is given.
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REFERENCES
Bass, R. F. (1985). Law of the iterated logarithm for set-indexed partial-sum processes with finite variance, Wahrsch. Verw. Geb. 70, 591–608.
Bickel, P. J., and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Stat. 42, 1656–1670.
Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields, Ann. Prob. 10, 1047–1050.
Chen, D. (1991). A uniform central limit theorem for nonuniform φ-mixing random fields, Ann. Probab. 19, 635–649.
Deo, Ch. M. (1975). A functional central limit theorem for stationary random fields, Ann. Prob. 3, 708–715.
Dudley, R. M. (1973). Sample functions of the Gaussian process, Ann. Prob. 1, 66–103.
Eberlein, E. (1979). An invariance principle for lattices of dependent random variables, Wahrsch. Verw. Geb. 50, 119–133.
Goldie, C. M., and Greenwood, P. E. (1986). Characterization of set-indexed Brownian motion and associated conditions for finite-dimensional convergence, Ann. Prob. 14, 803–816.
Goldie, C. M, and Greenwood, P. E. (1986). Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes, Ann. Prob. 14, 817–839.
Künsch, H. (1982). Decay of Correlations under Dobrushin uniqueness condition and its application, Commun. Math. Phys 84, 207–222.
Nahapetian, B. S. (1980). The central limit theorem for random fields with mixing conditions. In Dobrushin, R. L., and Sinai, Ya. G. (eds.), Multi-component System. Advances in Probability, Dekker, New York, pp. 531–548.
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Maltz, A.L. On the Central Limit Theorem for Nonuniform φ-Mixing Random Fields. Journal of Theoretical Probability 12, 643–660 (1999). https://doi.org/10.1023/A:1021619613916
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DOI: https://doi.org/10.1023/A:1021619613916