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.
"Limit Theorems for a Triangular Scheme of $U$-Statistics with Applications to Inter-Point Distances." Ann. Probab. 14 (4) 1347 - 1358, October, 1986. https://doi.org/10.1214/aop/1176992375