On asymptotic properties of bootstrap for AR(1) processes

We consider a first-order autoregressive process when the autoregressive parameter beta may vary over the entire real line. The standard bootstrap approximation to the sampling distribution of the least squares estimator of beta is shown to converge weakly to a random (i.e., nondegenerate) limit for the usual choice of the bootstrap sample size when beta equals 1 or -1. The bootstrap approximation, however, is asymptotically valid in probability, or even almost surely, for suitably selected resample sizes, whatever beta may be.