Limit Theorems for a Triangular Scheme of $U$-Statistics with Applications to Inter-Point Distances

Abstract

The asymptotic distribution of a "triangular" scheme of $U$-statistics is studied. Two limit theorems, applicable in different situations, are given. One theorem yields convergence to a normal distribution; the other includes Poisson limits and other limit laws. Applications to statistics based on small interpoint distances in a sample are given.