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.

翻译：Hoeffding不等式是概率论、统计学和机器学习中广泛应用的基础工具。本文针对可数状态空间上具有不变概率分布${\pi}$和${\mathcal{L}^{2}(\pi)}$-谱间隙${\lambda}(Q)$的不可约正常返连续时间马尔可夫链(CTMC)，建立了专门的Hoeffding不等式。具体地，对于取值区间为$[a,b]$且均值为$\pi(g)$的函数$g:E\to [a,b]$，给定$t,\varepsilon>0$，我们推导出以下不等式： \[ \mathbb{P}_{\pi}\left(\frac{1}{t} \int_{0}^{t} g\left(X_{s}\right)\mathrm{d}s-\pi (g) \geq \varepsilon \right) \leq \exp\left\{-\frac{{\lambda}(Q)t\varepsilon^2}{(b-a)^2} \right\},\] 该不等式可视为[J. Fan等, J. Mach. Learn. Res., 22(2022), pp. 6185-6219]中离散时间马尔可夫链(DTMC)的Hoeffding不等式在CTMC领域的推广。实现该不等式的关键分析在于利用骨架链技术与增广截断逼近方法。此外，我们还讨论了一般状态空间上跳跃过程的Hoeffding不等式。