Asymptotic bootstrap validity is usually understood as consistency of the dis-tribution of a bootstrap statistic, conditional on the data, for the unconditionallimit distribution of a statistic of interest. From this perspective, randomness ofthe limit bootstrap measure is regarded as a failure of the bootstrap. Nevertheless,apart from an unconditional limit distribution, a statistic of interest may possess ahost of (random) conditional limit distributions. This allows the understanding ofbootstrap validity to be widened, while maintaining the requirement of asymptoticcontrol over the frequency of correct inferences. First, we provide conditions forthe bootstrap to be asymptotically valid as a tool for conditional inference, in caseswhere a bootstrap distribution estimates consistently, in a sense weaker than thestandard weak convergence in probability, a conditional limit distribution of a sta-tistic. Second, we prove asymptotic bootstrap validity in a more basic, on-averagesense, in cases where the unconditional limit distribution of a statistic can be ob-tained by averaging a (random) limiting bootstrap distribution. Third, we applyour framework to several inference problems including functional CUSUM statis-tics, conditional Kolmogorov-Smirnov speci cation tests and tests for constancy ofparameters in dynamic econometric models.
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