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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

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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