Bootstrapping General Empirical Measures

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