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June 1999 A new mixing notion and functional central limit theorems for a sieve bootstrap in time series
Peter J. Bickel, Peter Bühlmann
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Bernoulli 5(3): 413-446 (June 1999).

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 , we approximate the underlying process by an autoregressive model with order p =p(n) , where p (n),p(n)=o(n) as the sample size n . Based on such a model, a bootstrap process { X t *} t 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 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.

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Peter J. Bickel. Peter Bühlmann. "A new mixing notion and functional central limit theorems for a sieve bootstrap in time series." Bernoulli 5 (3) 413 - 446, June 1999.

Information

Published: June 1999
First available in Project Euclid: 27 February 2007

zbMATH: 0954.62102
MathSciNet: MR1693612

Keywords: AR(∞) , ARMA , autoregressive approximation , bracketing , convex sets , linear process , MA(∞) , smooth bootstrap , stationary process , strong-mixing

Rights: Copyright © 1999 Bernoulli Society for Mathematical Statistics and Probability

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Vol.5 • No. 3 • June 1999
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