Inferring the Mixing Properties of a Stationary Ergodic Process From a Single Sample-Path

Abstract

We propose strongly consistent estimators of the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> 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 ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> 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 ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> 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.