Convergence of self-normalized generalizedU-statistics

Conditions are given for almost certain and distribution convergence of self-normalized generalizedU-statistics composed of random variables without particular probabilistic structure. The set of almost certain limit points of some classicalU-statistics is obtained. A variant of theU-statistic involving squares of some of the random variables is also treated. Applications include Martingale differences, stationary sequences, and the classical i.i.d. case where a Marcinkiewicz-Zygmund-type strong law is obtained.