Inferring the Mixing Properties of a Stationary Ergodic Process From a Single Sample-Path
Azadeh Khaleghi, Gábor Lugosi · IEEE Transactions on Information Theory · 2023
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.