  Hoeffding's inequality is a fundamental tool widely applied in probability
theory, statistics, and machine learning. In this paper, we establish
Hoeffding's inequalities specifically tailored for an irreducible and positive
recurrent continuous-time Markov chain (CTMC) on a countable state space with
the invariant probability distribution ${\pi}$ and an
$\mathcal{L}^{2}(\pi)$-spectral gap ${\lambda}(Q)$. More precisely, for a
function $g:E\to [a,b]$ with a mean $\pi(g)$, and given $t,\varepsilon>0$, we
derive the inequality \[ \mathbb{P}_{\pi}\left(\frac{1}{t} \int_{0}^{t}
g\left(X_{s}\right)\mathrm{d}s-\pi (g) \geq \varepsilon \right) \leq
\exp\left\{-\frac{{\lambda}(Q)t\varepsilon^2}{(b-a)^2} \right\}, \] which can
be viewed as a generalization of Hoeffding's inequality for discrete-time
Markov chains (DTMCs) presented in [J. Fan et al., J. Mach. Learn. Res.,
22(2022), pp. 6185-6219] to the realm of CTMCs. The key analysis enabling the
attainment of this inequality lies in the utilization of the techniques of
skeleton chains and augmented truncation approximations. Furthermore, we also
discuss Hoeffding's inequality for a jump process on a general state space.
