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Bootstrapping general empirical measures. (English) Zbl 0706.62017
It is proved that the central limit theorem for the bootstrapped empirical process indexed by a class of functions $${\mathcal F}$$ (satisfying certain measurability conditions) and based on a probability measure P holds a.s. if and only if the initial empirical process satisfies a central limit theorem ($${\mathcal F}\in CLT(P))$$ and $$\int F^ 2dP<\infty$$, where $$F=\sup_{f\in {\mathcal F}}| f|;$$ and it holds in probability if and only if $${\mathcal F}\in CLT(P)$$. The law of large numbers for the bootstrapped empirical process is also characterized.
Reviewer: A.J.Račkauskas

##### MSC:
 62E20 Asymptotic distribution theory in statistics 60F17 Functional limit theorems; invariance principles 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 62G30 Order statistics; empirical distribution functions
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