BLOCKWISE BOOTSTRAPPED EMPIRICAL PROCESS FOR STATIONARY-SEQUENCES

We apply the bootstrap for general stationary observations, proposed b y Kunsch, to the empirical process for p-dimensional random vectors. I t is known that the empirical process in the multivariate case converg es weakly to a certain Gaussian process. We show that the bootstrapped empirical process converges weakly to the same Gaussian process almos t surely assuming that the block length l for constructing bootstrap r eplicates satisfies l(n) = O(n(1/2 -epsilon) ), 0 < epsilon < 1/2, and l(n) --> infinity. An example where the multivariate setup arises are the robust GM-estimates in an autoregressive model. We prove the asym ptotic validity of the bootstrap approximation by showing that the fun ctional associated with the GM-estimates is Frechet-differentiable.