Mixing Property and Functional Central Limit Theorems for a Sieve

Mixing Property and Functional Central Limit Theorems for a Sieve

Report Number
440
Authors
Peter J. Bickel and Peter Bühlmann
Citation
Physics of the Earth and Planetary Interiors, December, 1996.
Abstract

We study a bootstrap method for stationary real-valued time series, which is based on the method of sieves. We restrict ourselves to autoregressive sieve bootstraps. Given a sample X_1,...,X_n from a linear process {X_t}_{t in Z}, we approximate the underlying process by an autoregressive model with order p=p(n), where p(n) tends to infinity, p(n)=o(n) as the sample size n tends to infinity. Based on such a model a bootstrap process {X_t^*}_{t in Z} is constructed from which one can draw samples of any size.

We give a novel result which says 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 the sieve bootstrap process. As an example we derive a functional central limit theorem under a bracketing condition.

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