Inferring the mixing properties of an ergodic process

Abstract

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.