A Stochastic Integral Representation for the Bootstrap of the Sample Mean

Abstract

Assume that F is the distribution function of a symmetric distribution in the domain of attraction of an α-stable distribution. Let X ₁, X ₂,... be a sequence of independent random variables distributed according to F. As Feller [3] shows, if 0 < α < 2, then there exist constants a _n > 0 such that n(1 - F(a _n y)) → y ^-α as n → ∞, and for such a _n S _n = a ^-1 _n (X ₁+...+ X _n ) converges in distribution to an α-stable random variable. If α = 2, then choose a _n such that a ^-1 _n (X ₁+...+ X _n ) converges in distribution to the standard normal.

Research partially supported by ONR Grant N000014-85-K-0150