AI 中文总结
针对单条轨迹下 $\phi$-混合系数扩展和的可估计性问题，构造了几乎必然收敛的确定性统计量，无需混合速率假设，并覆盖无穷和情形。
AI 中文摘要
Khaleghi 和 Lugosi 曾提出如下问题：对于一个实值离散时间平稳遍历过程，其 $\phi$-混合（一致混合）系数之和能否基于单条样本路径进行相合估计？我们构造了一个确定性的 Borel 统计量序列，使得对每个这样的过程，该序列几乎必然收敛到扩展和 $\sum_{m\geq1}\phi(m)$，包括当和为无穷时发散到 $+\infty$ 的情形。该估计器结合了有限二元柱（dyadic cylinders）与一个基于条件事件频率的趋于零的经验截断。当目标为有限时，$\alpha(m)\leq\phi(m)$ 为不断增长的有限类提供了可求和的自协方差控制，而稳定划分估计给出了上界。固定正概率见证与 Birkhoff 定理给出了下界，包括无穷目标的情形。该方法不要求任何混合速率或条件事件概率的已知正下界。
英文摘要
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 $\sum_{m\geq1}ϕ(m)$, 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, $α(m)\leqϕ(m)$ 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.
