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

TL;DR

This paper introduces a generalized notion of weak dependence called $$-weak dependence, which extends beyond mixing conditions, and demonstrates its application in analyzing autoregressive processes and their bootstrap counterparts.

Contribution

It defines the $$-weak dependence concept, shows how it applies to autoregressive processes, and provides probabilistic results for processes with this property.

Findings

Weak dependence can be derived from contraction properties.

Autoregressive processes exhibit $$-weak dependence.

Probabilistic results extend to processes with $$-weak dependence.

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.