Inferring the mixing properties of an ergodic process

TL;DR

This work tackles the problem of inferring the mixing structure of a stationary ergodic process by constructing strongly consistent estimators for the $\alpha$- and $\beta$-mixing coefficients and their $\ell_1$ norms from finite samples. It introduces cylinder-set based finite-dimensional approximations, proves convergence and concentration results (via Rio's covariance inequality), and develops two estimation regimes: strong consistency under summable $\ell_1$ norms and non-summable cases with weak/strong estimators. The authors further translate these estimators into strongly consistent goodness-of-fit tests for rate functions and for the norms, including a corollary giving a robust independence test. Overall, the paper provides a principled, nonparametric approach to validate mixing assumptions and detect independence in dependent time series without strong distributional assumptions, with implications for concentration inequalities and statistical testing under dependence.

Abstract

We propose strongly consistent estimators of the $\ell_1$ 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 $\ell_1$ norm of the sequence $\boldsymbolα$ of $α$-mixing coefficients of the process is bounded by a given threshold $γ\in [0,\infty)$ from the alternative hypothesis that $\left\lVert \boldsymbolα \right\rVert> γ$. An analogous goodness-of-fit test is proposed for the $\ell_1$ 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.

Theorems & Definitions (33)

• Lemma 1
• proof : Proof of Lemma \ref{['lem:dynk']}
• Proposition 1
• proof : Proof of Proposition \ref{['prop:phik']}
• Lemma 2: Rio RIO99
• Lemma 3
• proof : Proof of Lemma \ref{['lem:empf']}
• Lemma 4
• Proposition 2
• proof : Proof of Proposition \ref{['prop:phimb']}