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.