A class of \(U\)-statistics and asymptotic normality of the number of \(k\)- clusters (Q1206454)

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 \textit{S. R. Jammalamadaka} and \textit{S. Janson} [Ann. Probab. 14, 1347-1358 (1986; Zbl 0604.60023)] to general kernels and to general orders \(K>2\) of the kernel.