Blockwise Bootstrapped Empirical Process for Stationary Sequences

Abstract

We apply the bootstrap for general stationary observations, proposed by Kunsch, to the empirical process for $p$-dimensional random vectors. It is known that the empirical process in the multivariate case converges weakly to a certain Gaussian process. We show that the bootstrapped empirical process converges weakly to the same Gaussian process almost surely, assuming that the block length $l$ for constructing bootstrap replicates satisfies $l(n) = O(n^{1/2-\varepsilon}), 0 < \varepsilon < \frac{1}{2}$, and $l(n) \rightarrow \infty$. An example where the multivariate setup arises are the robust GM-estimates in an autoregressive model. We prove the asymptotic validity of the bootstrap approximation by showing that the functional associated with the GM-estimates is Frechet-differentiable.