Unconditional Glivenko-Cantelli-type theorems and weak laws of large numbers for bootstrap

We first give some Glivenko-Cantelli-type theorems for the bootstrap. We prove almost sure convergence and convergence in probability. The results involve conditions on the resampling size which are not far from being (and often are) optimal. We next analyze the weak law of large numbers for the bootstrap mean. The conditions needed for these weak laws are related to the Glivenko-Cantelli statements on convergence in probability. This relation makes it possible to obtain weak laws in very general situations. In particular our techniques work with smoothed boostrap, and even lead to a proof of the strong law for the mean when kernel estimators are used.