Statistical tests for uniformity of distribution and independence of vectors based on pairwise distances

The problem of testing statistical hypotheses on the uniformity of distribution and independence of random vectors is interesting from theoretical and practical viewpoints. Many authors had proposed various approaches to solving it. In some tests for the hypotheses on the uniformity of distributions of random vectors in a multidimensional region or on their independence some functions of pairwise distances between sample points are used as statistics. The paper contains a brief review of such tests. A new test for the hypothesis on the uniformity of distribution based on pairwise distances is proposed.