A residual-based bootstrap for functional autoregressions

Abstract

We consider the residual-based or naive bootstrap for functional autoregressions of order 1 and prove that it is asymptotically valid for, e.g., the sample mean and for empirical covariance operator estimates. As a crucial auxiliary result, we also show that the empirical distribution of the centered sample innovations converges to the distribution of the innovations with respect to the Mallows metric.