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On the spectral density and asymptotic normality of weakly dependent random fields. (English) Zbl 0787.60059

Let \(X=(X_ k,k\in\mathbb{Z}^ d)\) be a centered complex weakly stationary random fields. For any disjoint \({\mathcal S},{\mathcal D}\subset\mathbb{Z}^ d\) define \(\sup{EV\overline W\over(\| V\|_ 2\cdot\| W\|_ 2}=r(S,{\mathcal D})\) where the supremum is taken over all pairs of random variables \(V=\sum_{k\in S^*}a_ kX_ k\), \(W=\sum_{k\in{\mathcal D}^*}b_ kX_ k\), where \({\mathcal S}^*\) and \({\mathcal D}^*\) are finite subset of \({\mathcal S}\) and \({\mathcal D}\), and \(a_ k\) and \(b_ k\) are complex numbers. For every real number \(s\geq 1\), define \(r^*(s)=\sup r({\mathcal S},{\mathcal D})\), where the supremum is taken over all pairs of nonempty disjoint subsets \({\mathcal S},{\mathcal D}\subset\mathbb{Z}^ d:\text{dist}({\mathcal S},{\mathcal D})\geq s\).
Theorem 1. If \(r^*(s)\to 0\) as \(s\to\infty\), then \(X\) has a continuous spectral density.
Theorem 2. The following two statements are equivalent: (1) \(r^*(1)<1\) and \(r^*(s)\to 0\) as \(s\to\infty\); (2) \(X\) has a continuous spectral density.
The central limit theorem is proved. No mixing rate is assumed.
Reviewer: N.Leonenko (Kiev)

MSC:

60G60 Random fields
60G10 Stationary stochastic processes
60F05 Central limit and other weak theorems
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[1] Bergh, J., and Löfström, J. (1976).Interpolation Spaces, Springer, New York. · Zbl 0344.46071
[2] Bradley, R. C. (1987). The central limit question under ?-mixing.Rocky Mountain J. Math. 17, 95-114. · Zbl 0646.60027
[3] Bradley, R. C. (1988). A central limit theorem for stationary ?-mixing sequences with infinite variance.Ann. Prob. 16, 313-332. · Zbl 0643.60018
[4] Bradley, R. C., and Bryc, W. (1985). Multilinear forms and measures of dependence between random variables.J. Multivar. Anal. 16, 335-367. · Zbl 0586.62086
[5] Dvoretzky, A. (1972). Asymptotic normality for sums of dependent random variables.Sixth Berkeley Symp. Math. Stat. Prob. 2, 513-535.
[6] Goldie, C. M., and Greenwood, P. (1986). Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes.Ann. Prob. 14, 817-839. · Zbl 0604.60032
[7] Goldie, C. M., and Morrow, G. J. (1986). Central limit questions for random fields. In Eberlein, E., and Taqqu, M. S., (eds.).Dependence in Probability and Statistics, Progress in Probability and Statistics, Vol. 11, Birkhäuser, Boston, pp. 275-289. · Zbl 0605.60029
[8] Gorodetskii, V. V. (1984). The central limit theorem and an invariance principle for weakly dependent random fields.Soviet Math. Dokl. 29, 529-532. · Zbl 0595.60025
[9] Ibragimov, I. A. (1975). A note on the central limit theorem for dependent random variables.Theor. Prob. Appl. 20, 135-141. · Zbl 0335.60023
[10] Ibragimov, I. A., and Linnik, Yu. V. (1971).Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen. · Zbl 0219.60027
[11] Ibragimov, I. A., and Rozanov, Y. A. (1978).Gaussian Random Processes, Springer, New York.
[12] Peligrad, M. (1986). Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (a survey). In Eberlein, E., and Taqqu, M. S. (eds.),Dependence in Probability and Statistics, Progress in Probability and Statistics, Vol. 11, Birkhäuser, Boston, pp. 193-223. · Zbl 0603.60022
[13] Rosenblatt, M. (1985).Stationary Sequences and Random Fields, Birkhäuser, Boston. · Zbl 0597.62095
[14] Sarason, D. (1972). An addendum to ?Past and Future?.Math. Scand. 30, 62-64. · Zbl 0266.60023
[15] Shao, Q. (1986).An Invariance Principle for Stationary ?-Mixing Sequences with Infinite Variance. Report, Department of Mathematics, Hangzhou University, Hangzhou, Peoples Republic of China.
[16] Withers, C. S. (1981). Central limit theorems for dependent random variables.Z. Wahrsch. verw. Gebiete 57, 509-534. · Zbl 0451.60027
[17] Zhurbenko, I. G. (1986).The Spectral Analysis of Time Series, North-Holland, Amsterdam. · Zbl 0593.62095
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