Laws of large numbers for bootstrappedU-statistics

For the bootstrapped mean, a strong law of large numbers is obtained under the assumption of finiteness of the rth moment, for some r>1, and a weak law of large numbers is obtained under the finiteness of the first moment. The results are then extended to bootstrapped U-statistics under parallel conditions. Stochastic convergence of the jackknifed estimator of the variance of a bootstrapped U-statistic is proved. The asymptotic normality of the bootstrapped pivot and the bias of the bootstrapped U-statistic are indicated.