Abstract
This paper generalizes results by Bradley.(3) Suppose that for 1=1,2,...X 1 k :k ∈ ℤd is a centered, weakly stationary ρ*-mixing random field, and suppose liml→∞ Cov(X 10 ,x 1 k ) exists, anyk ∈ ℤd. Then the successive spectral densities converge uniformly to a continuous function. For a sequence of strictly stationary random fields that are uniformly ρ*-mixing and satisfy a indeberg condition, a CLT is proved for sequences of sums from the fields. This result is then applied: given a centered strictly stationary ρ*-mixing random field whose probability density and joint densities are continuous, then a kernel estimator for the probability density obeys the CLT.
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References
Bradley, R. C. (1983). Asymptotic normality of some kernel-type estimators of probability density.Stat. and Prob. Letters 1, 295–300.
Bradley, R. C. (1986). Basic properties of strong mixing conditions. In Eberlein, E., and Taqqu, M. S. (eds.),Dependence in Probability and Statistics: A Survey of Recent Results, Progress in Probability and Statistics 11, 165–192.
Bradley, R. C. (1992). On the spectral density and asymptotic normality of dependence between random variables.J. Th. Prob. 5, 355–373.
Bradley, R. C. (1993). Some examples of mixing random fields.Rocky Mtn. Jr. of Math. 23, 495–519.
Bradley, R. C. Equivalent mixing conditions for random fields.Ann. Prob. (to appear).
Falk, M. (1984). On the convergence of spectral densities of arrays of weakly stationary processes.Ann. Prob. 12, 918–921.
Ibragimov, I. A., and Linnik, Yu. V. (1971).Independent and Stationary Sequences of Random Variables, Walters-Noordhoff, Groningen.
Rosenblatt, M. (1985).Stationary Sequences and Random Fields, Birkhäuser, Boston.
Tran, L. T. (1990). Kernel density estimation in random fields.J. Multivariate Anal. 34, 37–53.
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Miller, C. Three theorems on ρ* random fields. J Theor Probab 7, 867–882 (1994). https://doi.org/10.1007/BF02214377
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DOI: https://doi.org/10.1007/BF02214377