Efron's "bootstrap" method of distribution approximation is shown to be asymptotically valid in a large number of situations, including $t$-statistics, the empirical and quantile processes, and von Mises functionals. Some counter-examples are also given, to show that the approximation does not always succeed.
"Some Asymptotic Theory for the Bootstrap." Ann. Statist. 9 (6) 1196 - 1217, November, 1981. https://doi.org/10.1214/aos/1176345637