Functional dependence and synchronous coupling in ergodic autoregressions

Abstract

Functional dependence measures have become an important tool in the analysis of nonlinear time series and are typically formulated with respect to a given innovation representation of the process. This note points out that the probability space on which such representations yield the expected memory loss properties may not always coincide with the natural dynamical probability space of the model. We exhibit classes of uniformly ergodic autoregressive processes for which the behavior of the natural innovation coupling undergoes a qualitative transition as the model parameter varies. For this family of models, this transition coincides with a change in the sign of an associated Lyapunov exponent. In particular, a positive Lyapunov exponent may prevent the forgetting of initial perturbations along trajectories driven by the same innovations, despite uniform ergodicity of the associated Markov chain. These observations highlight the importance of carefully specifying the underlying probability space when interpreting or applying functional dependence measures.