In the non-lattice case it is shown that the bootstrap approximation of the distribution of the standardized sample mean is asymptotically more accurate than approximation by the limiting normal distribution. The exact convergence rate of the bootstrap approximation of the distributions of sample quantiles is obtained. A few other convergence rates regarding the bootstrap method are also studied.
"On the Asymptotic Accuracy of Efron's Bootstrap." Ann. Statist. 9 (6) 1187 - 1195, November, 1981. https://doi.org/10.1214/aos/1176345636