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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 20, No. 1, pp. 93-104 (2004)
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A central limit theorem for random fields

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István Fazekas and Alexey Chuprunov

University of Debrecen, Hungary; Kazan State University, Russia

**Abstract:**
A central limit theorem is proved for $\alpha$-mixing random fields. The sets of locations where the random field is observed become more and more dense in an increasing sequence of domains. The central limit theorem concerns these observations. The limit theorem is applied to obtain asymptotic normality of kernel type density estimators. It turns out that in our setting the covariance structure of the limiting normal distribution can be a combination of those of the continuous parameter and the discrete parameter cases.

**Keywords:** Central limit theorem, random field, $\alpha$-mixing, infill asymptotics, increasing domain asymptotics, density estimator, asymptotic normality of estimators.

**Classification (MSC2000):** 60F05; 62M30

**Full text of the article:**

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© 2004 ELibM and
FIZ Karlsruhe / Zentralblatt MATH
for the EMIS Electronic Edition
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