Poisson limits for \(U\)-statistics. (Q1766076)

In the first part of the paper point processes of structure like a \(U\)-statistic with a symmetric non-negative kernel are considered and their convergence to a Poisson process is established under conditions of regularity and asymptotic independence. The result is then utilized to obtain \(\alpha \)-stable limits for \(U\)-statistics with heavy right-tailed kernels for stable indices \(\alpha \in (0,1) \) and \(\alpha \in (1,2).\) Estimators of the index of regular variation (correlation dimension) like F. Takens and B. M. Hill estimators are considered and their asymptotic properties are studied.