We establish Bernstein's inequalities for functions of general (general-state-space and possibly non-reversible) Markov chains. These inequalities achieve sharp variance proxies and encompass the classical Bernstein inequality for independent random variables as special cases. The key analysis lies in bounding the operator norm of a perturbed Markov transition kernel by the exponential of sum of two convex functions. One coincides with what delivers the classical Bernstein inequality, and the other reflects the influence of the Markov dependence. A convex analysis on these two functions then derives our Bernstein inequalities. As applications, we apply our Bernstein inequalities to the Markov chain Monte Carlo integral estimation problem and the robust mean estimation problem with Markov-dependent samples, and achieve tight deviation bounds that previous inequalities can not.

翻译：我们针对一般（状态空间一般且可能不可逆）马尔可夫链的函数建立了伯恩斯坦不等式。这些不等式实现了最优方差代理，并将经典独立随机变量的伯恩斯坦不等式作为特例纳入其中。核心分析在于通过两个凸函数之和的指数来界定扰动马尔可夫转移核的算子范数：其中一个凸函数与经典伯恩斯坦不等式的构造一致，另一个则反映了马尔可夫依赖性的影响。通过对这两个函数的凸分析，我们推导出了新的伯恩斯坦不等式。作为应用，我们将这些不等式应用于马尔可夫链蒙特卡洛积分估计问题及马尔可夫依赖样本的稳健均值估计问题，获得了以往不等式无法实现的紧致偏差界。