Bootstrapping general empirical measures (Q918584)

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.