The notion of $\psi$-weak dependence and its applications to bootstrapping time series

Abstract

We give an introduction to a notion of weak dependence which is more general than mixing and allows to treat for example processes driven by discrete innovations as they appear with time series bootstrap. As a typical example, we analyze autoregressive processes and their bootstrap analogues in detail and show how weak dependence can be easily derived from a contraction property of the process. Furthermore, we provide an overview of classes of processes possessing the property of weak dependence and describe important probabilistic results under such an assumption.

arXiv commentPublished in at http://dx.doi.org/10.1214/06-PS086 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions
• An invariance principle for weakly dependent stationary general models
• Goodness-of-fit tests for Markovian time series models: Central limit theory and bootstrap approximations
• Strong invariance principle for dependent random fields

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