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 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 \(g:E\rightarrow [a,b]\) with a mean \(\pi (g)\) , and given \(t,\varepsilon >0\) , we derive the inequality \(\begin{aligned} \mathbb {P}_{\pi }\left( \frac{1}{t} \int _{0}^{t} g\left( X_{s}\right) \textrm{d}s-\pi (g) \ge \varepsilon \right) \le \exp \left\{ -\frac{{\lambda }(Q)t\varepsilon ^2}{(b-a)^2} \right\} , \end{aligned}\) which can be viewed as a generalization of Hoeffding’s inequality for discrete-time Markov chains presented in Fan et al. (J. Mach. Learn. Res., 22(139), 6185–6219 2021) to the realm of continuous-time Markov chains. 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.
