Asymptotic normality of a random point field characteristic

摘要

In the paper a sequence of bounded regions containing n independent identically and uniformly on Dndistributed points is considered. It is assumed that the d#x2013;dimensional volume v(Dn) is asymptotically proportional to n. Under these conditions it is shown that the number of pairs of points within a distance r0 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#x2013;statistics are insufficient. The obtained result is applied for testing the hypothesis of randomness