Bootstrapping General Empirical Measures

Abstract

It is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions F and based on a probability measure P holds a.s. if and only if F ∈ CLT(P) and $\int F^2 dP < \infty$ , where $F = \sup_{f \in \mathscr{F}}|f|$ , and it holds in probability if and only if F ∈ CLT(P). Thus, for a large class of statistics, no local uniformity of the CLT (about P) is needed for the bootstrap to work. Consistency of the bootstrap (the bootstrapped law of large numbers) is also characterized. (These results are proved under certain weak measurability assumptions on F.)