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Strongly Consistent Estimation of the Extended ℓ1-Sum of ϕ-Mixing Coefficients from a Single Trajectory

  • Senhan Yao
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Abstract

Khaleghi and Lugosi asked whether the sum of the ϕ-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 ∑m≥1ϕ(m), including divergence to +∞ 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, α(m)≤ϕ(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.

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Cite

BibTeXRIS
BibTeX
@misc{yao2026strongly,
  title = {{Strongly Consistent Estimation of the Extended $\ell_1$-Sum of $ϕ$-Mixing Coefficients from a Single Trajectory}},
  author = {Senhan Yao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26232},
  eprint = {2609.26232},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26232},
}
RIS
TY  - UNPB
TI  - Strongly Consistent Estimation of the Extended ℓ_1-Sum of ϕ-Mixing Coefficients from a Single Trajectory
AU  - Senhan Yao
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.26232
UR  - https://arxiv.org/abs/2609.26232
AB  - Khaleghi and Lugosi asked whether the sum of the ϕ-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 ∑_m≥1ϕ(m), including divergence to +∞ 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, α(m)≤ϕ(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.
ER  -