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

Abstract

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 $\{X_t{\}}_{t\in \mathbb{Z}}$, we approximate the underlying process by an autoregressive model with order $p=p\left(n\right)$, where $p\left(n\right)\to \mathrm{\infty },p\left(n\right)=o\left(n\right)$ as the sample size $n\to \mathrm{\infty }$. Based on such a model, a bootstrap process $\{X_t^{*}{\}}_{t\in \mathbb{Z}}$ is constructed from which one can draw samples of any size.

We show that, with high probability, such a sieve bootstrap process $\{X_t^{*}{\}}_{t\in \mathbb{Z}}$ 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.