Asymptotic Properties of the Bootstrap for Heavy-Tailed Distributions

Abstract

We establish necessary and sufficient conditions for convergence of the distribution function of a bootstrapped mean, suitably normalized. It turns out that for convergence to occur, the sampling distribution must either be in the domain of attraction of the normal distribution or have slowly varying tails. In the first case the limit is normal; in the latter, Poisson. Between these two extremes of light tails and extremely heavy tails, the bootstrap distribution function of the mean does not converge in probability to a nondegenerate limit. However, it may converge in distribution. We show that when there is no convergence in probability, a small number of extreme sample values determine behaviour of the bootstrap distribution function. This result is developed and used to interpret recent work of Athreya.