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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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