Stein’s Method for the Bootstrap Dedicated to Charles Stein on the occasion of his 80th birthday

Stein’s method proves weak-convergence results through approximations to expectations without direct use of characteristic functions, allowing it to be used in complex problems with dependence. In this work we show how consistency can be proved and even some error terms provided for any bootstrap with exchangeable weights using Stein’s method. We say that the bootstrap works if the distance between the bootstrap empirical measure and a Gaussian measure centred around the true empirical measure, or the true mean measure, tends to zero as sample size tends to infinity. Our results also provide an explicit error bound for the difference to Gaussianity of the bootstrap for any finite sample size. Many of the results themselves are known, see for instance Bickel and Freedman(1981), Singh(1981) for the consistency results in the multinomial case or Praestgaard and Wellner(1993) for the case of exchangeable weights. However this paper proposes a new way of bounding error terms for the bootstrap that does not rely on Edgeworth expansions as does the other theoretical work on convergence rates to date. In the independent case see (Hall, 1986, 1988b,a; Hall and Martin, 1991; Hall, 1992). Examples of proofs for dependent variables using Edgeworth expansions can be found in Lahiri (1995) and the book by Politis et al. (1999). Instead of comparing two distributions directly, in this approach we compare their Stein operators on certain test functions and the expectation of their difference is used to bound the actual distance between distributions. After defining the operators and notations, we start by the simple case of the consistency of the bootstrap distribution for the mean following Stein’s proof of the central limit theorem closely. We