On the convergence of \(U\)-statistics with stable limit distribution (Q1209882)

Weak sufficient conditions are given to ensure the convergence of the distribution of suitably normalized \(U\)-statistics to an \(\alpha\)-stable distribution, \(1<\alpha\leq 2\). The proofs use Hoeffding's decomposition for \(U\)-statistics. For a subset of cases covered by the above result a rate of convergence is given which is of the same order as the rate obtained for similar stable limit results for sums of independent and identically distributed random variables.