A new mixing notion and functional central limit theorems for a sieve bootstrap in time series

We study a bootstrap method for stationary real-valued time series, which is based on the sieve of autoregressive: processes. Given a sample X-1,...,X-n from a linear process {X-t}(t is an element of Z), We approximate the underlying process by an autoregressive model with order p = p(n),where p(n) --> infinity p(n) = o(n) as the sample size n --> infinity. Based on such a model, a bootstrap process {X-t*}(t is an element of Z) is constructed from which one can draw samples of any size. We show that, with high probability, such a sieve bootstrap process (X-t*)(t is an element of Z) satisfies a new type of mixing condition. This implies that many results for stationary mixing sequences carry over to the sieve bootstrap process. As an example we derive a functional central limit theorem under a bracketing condition.