Inferring the mixing properties of an ergodic process

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 $leftlVert boldsymbol{alpha} rightrVert> gamma$. 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.