Bootstrap and asymptotic normality

Abstract

In this chapter consistency of bootstrap is compared with asymptotic normality. This is done for linear statistics of n i. i. d. observations. It is shown that bootstrap works asymptotically under the same assumptions as a normal approximation with estimated variance (Theorem 1). This result is extended to the case of independent but not necessarily identically distributed observations (Theorem 2). Furthermore, bootstrap with a Poisson random sample size is considered. This bootstrap procedure is a special case of a class of resampling plans called wild bootstrap which have been proposed for the non i. i. d. case. We show that bootstrap works as long as the same holds for wild bootstrap (Theorem 3). The poissonisation is also a central tool for proving that consistency of bootstrap implies asymptotic normality.