A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. (English) Zbl 0954.62102

Bickel, Peter J.; Bühlmann, Peter

Bernoulli 5, No. 3, 413-446 (1999).

A bootstrap method for stationary real-valued time series based on the sieves of autoregressive processes is studied. Given a sample \(X_1,\dots,X_n\) from a linear process \((X_t)_{t\in Z}\), the underlying process is approximated by an autoregressive model of order \(p=p(n)\), where \(p(n)\to\infty, p(n)=o(n)\), as the sample size \(n\to\infty\). Based on such a model, a bootstrap process \((X^*_t)_{t\in Z}\) is constructed from which one can draw samples of any size. It is shown that, with high probability, such a sieve bootstrap process \((X^*_t)_{t\in Z}\) satisfies a new type of mixing condition. This implies that many results for stationary mixing sequences carry over to sieve bootstrap processes. As an example, a functional central limit theorem under bracketing condition is derived.

Reviewer: Yurii Lin’kov (Donetsk)

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F17	Functional limit theorems; invariance principles
62F40	Bootstrap, jackknife and other resampling methods

Keywords:

AR(\(\infty\)); ARMA; autoregressive approximation; linear processes; MA(\(\infty\)); smooth bootstrap; strong-mixing