This paper gives new proofs for many known results about the convergence in law of the bootstrap distribution to the true distribution of smooth statistics, whether the samples studied come from independent realizations of a random variable or dependent realizations with weak dependence. Moreover it suggests a novel bootstrap procedure and provides a proof that this new bootstrap works under uniform local dependence. The techniques employed are based on Stein’s method for empirical processes as developed by Reinert (1994). A weak law of large numbers for empirical measures via Stein’s method and applications. The last section provides some simulations and applications for which the relevant matlab functions are available from the first author.
Digital Object Identifier: 10.1214/lnms/1196283802