Inferring the Mixing Properties of a Stationary Ergodic Process From a Single Sample-Path

We propose strongly consistent estimators of the $\ell _{1}$ norm of the sequence of $\alpha $ -mixing (respectively $\beta $ -mixing) coefficients of a stationary ergodic process. We further provide strongly consistent estimators of individual $\alpha $ -mixing (respectively $\beta $ -mixing) coefficients for a subclass of stationary $\alpha $ -mixing (respectively $\beta $ -mixing) processes with summable sequences of mixing coefficients. The estimators are in turn used to develop strongly consistent goodness-of-fit hypothesis tests. In particular, we develop hypothesis tests to determine whether, under the same summability assumption, the $\alpha $ -mixing (respectively $\beta $ -mixing) coefficients of a process are upper bounded by a given rate function. Moreover, given a sample generated by a (not necessarily mixing) stationary ergodic process, we provide a consistent test to discern the null hypothesis that the $\ell _{1}$ norm of the sequence $\boldsymbol {\alpha }$ of $\alpha $ -mixing coefficients of the process is bounded by a given threshold $\gamma \in [0,\infty$ ) from the alternative hypothesis that $\left \lVert{ \boldsymbol {\alpha }}\right \rVert > \gamma $ . An analogous goodness-of-fit test is proposed for the $\ell _{1}$ norm of the sequence of $\beta $ -mixing coefficients of a stationary ergodic process. Moreover, the procedure gives rise to an asymptotically consistent test for independence.