Asymptotic normality of a random point field characteristic

Abstract

In the paper a sequence of bounded regions containing n independent identically and uniformly on D_n distributed points is considered. It is assumed that the d–dimensional volume v(D_n) is asymptotically proportional to n. Under these conditions it is shown that the number of pairs of points within a distance r>0 of each other is asymptotically normally distributed. For proving this among other things a lemma of BOLTHAUSEN is used, whereas even strong estimates for U–statistics are insufficient. The obtained result is applied for testing the hypothesis of randomness