Strong law of large numbers for weighted UU-statistics: Application to incomplete UU-statistics

In this exposition we investigate almost sure convergence for weighted UU-statistics. This strong law of large numbers is established for a class of divergent series of non-negative weights when the product of the kernel of the underlying weighted UU-statistics and its logarithm is integrable. This result is used to establish the almost sure validity of incomplete UU-statistics. Also, a method of constructing incomplete UU-statistics of order 2 is introduced and its almost sure validity is established.