Approximately mixing time series

In this note, we present the new concept of approximate mixing for random variables on metric spaces. Approximate mixing is characterized by two constants $ϵ,\delta \ge 0$, where $ϵ$ is the mixing coefficient and $\delta$ is a slack variable. In the case $\delta =0$, approximate mixing reduces to classical $\beta$-mixing. For positive slack, $\delta >0$, it becomes more general than traditional mixing assumptions, including important time series such as autoregressive processes on Hilbert spaces, that are generally not mixing. We prove that under approximate mixing analogous covariance inequalities hold as in the mixing case. We use these results to prove a central limit theorem for non-stationary time series on Hilbert spaces, which has potential applications in functional data analysis.