  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.
