We propose methods to estimate the individual $\beta$-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path $X_0,X_1, \dots,X_n$. Under standard smoothness conditions on the densities, namely, that the joint density of the pair $(X_0,X_m)$ for each $m$ lies in a Besov space $B^s_{1,\infty}(\mathbb R^2)$ for some known $s>0$, we obtain a rate of convergence of order $\mathcal{O}(\log(n) n^{-[s]/(2[s]+2)})$ for the expected error of our estimator in this case\footnote{We use $[s]$ to denote the integer part of the decomposition $s=[s]+\{s\}$ of $s \in (0,\infty)$ into an integer term and a {\em strictly positive} remainder term $\{s\} \in (0,1]$.}. We complement this result with a high-probability bound on the estimation error, and further obtain analogues of these bounds in the case where the state-space is finite. Naturally no density assumptions are required in this setting; the expected error rate is shown to be of order $\mathcal O(\log(n) n^{-1/2})$.

翻译：本文提出了一种从单条样本路径 $X_0,X_1,\dots,X_n$ 估计实值几何遍历马尔可夫过程各 $\beta$-混合系数的方法。在密度函数满足标准光滑性条件（即对每个 $m$，$(X_0,X_m)$ 的联合密度位于已知 $s>0$ 的 Besov 空间 $B^s_{1,\infty}(\mathbb R^2)$ 中）的情况下，我们得到了该情形下估计量期望误差的收敛速度为 $\mathcal{O}(\log(n) n^{-[s]/(2[s]+2)})$ 阶\footnote{此处 $[s]$ 表示将 $s \in (0,\infty)$ 分解为整数部分与{\em 严格正}余项 $\{s\} \in (0,1]$ 后，分解式 $s=[s]+\{s\}$ 的整数部分。}。我们进一步给出了该估计误差的高概率界，并将其推广至状态空间有限的情形。在此设定下无需密度假设；期望误差率被证明为 $\mathcal O(\log(n) n^{-1/2})$ 阶。