Estimating $\beta$-mixing coefficients

TL;DR

This paper introduces an estimator for the $eta$-mixing rate in time series data, providing a way to empirically assess the mixing assumptions often used in statistical learning, with proven consistency from a single sample.

Contribution

It proposes the first estimator for $eta$-mixing rates that is consistent based on a single stationary sample path, addressing a gap in testing mixing assumptions.

Findings

Estimator is $L_1$-risk consistent.

Provides a practical method to estimate mixing rates.

Addresses a key gap in statistical learning for time series.

Abstract

The literature on statistical learning for time series assumes the asymptotic independence or ``mixing' of the data-generating process. These mixing assumptions are never tested, nor are there methods for estimating mixing rates from data. We give an estimator for the $\beta$-mixing rate based on a single stationary sample path and show it is $L_1$-risk consistent.