Some Converse Limit Theorems for Exchangeable Bootstraps

The bootstrap Glivenko-Cantelli and bootstrap Donsker theorems of Giné and Zinn (1990) contain both necessary and sufficient conditions for the asymptotic validity of Efron’s nonparametric bootstrap. In the more general case of exchangeably weighted bootstraps, Praestgaard and Wellner (1993) and Van der Vaart and Wellner (1996) give analogues of the sufficiency half of the Theorems of Giné and Zinn (1990), but did not address the corresponding necessity parts of the theorems. Here we establish a new lower bound for exchangeably weighted processes and show that the necessity half of the a.s. bootstrap Glivenko-Cantelli theorem holds for exchangeably weighted bootstraps. We also make some progress toward the conjectured necessity parts of the bootstrap Donsker theorems with exchangeable weights.