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

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.

### MSC:

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