  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 $\mathcal
O(\log(n) n^{-1/2})$.
