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A class of $$U$$-statistics and asymptotic normality of the number of $$k$$- clusters. (English) Zbl 0764.60025
The authors prove a central limit theorem for a class of $$U$$-statistics whose kernel depends on the sample size and for which the projection method may fail. As an application they derive the asymptotic normality of the number of Poisson $$K$$-clusters in a cube of increasing size in $$R^ d$$. In this process they also extend earlier results of S. R. Jammalamadaka and S. Janson [Ann. Probab. 14, 1347-1358 (1986; Zbl 0604.60023)] to general kernels and to general orders $$K>2$$ of the kernel.

##### MSC:
 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics
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##### References:
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