Inferring the mixing properties of an ergodic process

Khaleghi, Azadeh and Lugosi, Gabor (2021) Inferring the mixing properties of an ergodic process. arXiv.org.

Abstract

We propose strongly consistent estimators of the ℓ1 norm of the sequence of α-mixing (respectively β-mixing) coefficients of a stationary ergodic process. We further provide strongly consistent estimators of individual α-mixing (respectively β-mixing) coefficients for a subclass of stationary α-mixing (respectively β-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 α-mixing (respectively β-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 ℓ1 norm of the sequence α of α-mixing coefficients of the process is bounded by a given threshold γ∈[0,∞) from the alternative hypothesis that ‖α‖>γ. An analogous goodness-of-fit test is proposed for the ℓ1 norm of the sequence of β-mixing coefficients of a stationary ergodic process. Moreover, the procedure gives rise to an asymptotically consistent test for independence.