In this paper, we establish novel deviation bounds for additive functionals of geometrically ergodic Markov chains similar to Rosenthal and Bernstein 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. More precisely, we establish explicit bounds that are linked to the constants from the martingale version of the Rosenthal inequality, as well as the constants that characterize the mixing properties of the underlying Markov kernel. Finally, our proof technique is, up to our knowledge, new and based on a recurrent application of the Poisson decomposition.

翻译：本文针对几何遍历马尔可夫链的可加泛函，建立了类似于独立随机变量和的Rosenthal与Bernstein不等式的新型偏差界。我们特别关注所得到的界对相应链混合时间的依赖关系。更精确地说，我们建立了显式界，这些界既与Rosenthal不等式的鞅版本中的常数相关，也与刻画底层马尔可夫核混合性质的常数相关。最后，据我们所知，本文的证明方法是全新的，基于泊松分解的递归应用。