Strongly Consistent Estimation of the Extended ${\ell }_1$-Sum of $\varphi$-Mixing Coefficients from a Single Trajectory

Abstract

Khaleghi and Lugosi asked whether the sum of the $\varphi$-mixing (uniform-mixing) coefficients of a real-valued discrete-time stationary ergodic process can be consistently estimated from a single sample path. We construct a deterministic sequence of Borel statistics that, for every such process, converges almost surely to the extended sum ${\sum }_{m\ge 1}\varphi (m)$, including divergence to $+\infty$ when the sum is infinite. The estimator combines finite dyadic cylinders with a vanishing empirical cutoff on conditioning-event frequencies. When the target is finite, $\alpha (m)\le \varphi (m)$ supplies summable covariance control for the growing finite classes, and a stable-division estimate yields the upper bound. Fixed positive-probability witnesses and Birkhoff's theorem give the lower bound, including the infinite-target case. No mixing rate or known positive lower bound on conditioning-event probabilities is required.