In this paper, we establish novel deviation bounds for additive functionals of geometrically ergodic Markov chains similar to Rosenthal and Bernstein-type inequalities for sums of independent random variables. We pay special attention to the dependence of our bounds on the mixing time of the corresponding chain. Our proof technique is, as far as we know, new and based on the recurrent application of the Poisson decomposition. We relate the constants appearing in our moment bounds to the constants from the martingale version of the Rosenthal inequality and show an explicit dependence on the parameters of the underlying Markov kernel.

翻译：本文针对几何遍历马尔可夫链的可加泛函，建立了与独立随机变量和的Rosenthal型和Bernstein型不等式类似的偏差界。我们特别关注所得边界对相应链的混合时间的依赖性。据我们所知，我们的证明方法是全新的，基于泊松分解的递归应用。我们将矩边界中出现的常数与Rosenthal不等式的鞅版本的常数相关联，并展示了它们对底层马尔可夫核参数的显式依赖性。