We introduce a novel concept of convergence for Markovian processes within Orlicz spaces, extending beyond the conventional approach associated with $L_p$ spaces. After showing that Markovian operators are contractive in Orlicz spaces, our key technical contribution is an upper bound on their contraction coefficient, which admits a closed-form expression. The bound is tight in some settings, and it recovers well-known results, such as the connection between contraction and ergodicity, ultra-mixing and Doeblin's minorisation. Specialising our approach to $L_p$ spaces leads to a significant improvement upon classical Riesz-Thorin's interpolation methods. Furthermore, by exploiting the flexibility offered by Orlicz spaces, we can tackle settings where the stationary distribution is heavy-tailed, a severely under-studied setup. As an application of the framework put forward in the paper, we introduce tighter bounds on the mixing time of Markovian processes, better exponential concentration bounds for MCMC methods, and better lower bounds on the burn-in period. To conclude, we show how our results can be used to prove the concentration of measure phenomenon for a sequence of Markovian random variables.

翻译：我们在Orlicz空间中引入了一种新的马尔可夫过程收敛概念，拓展了与$L_p$空间相关的传统方法。在证明马尔可夫算子在Orlicz空间中具有压缩性后，我们的核心技术贡献在于给出了其压缩系数的上界，该上界具有闭式表达式。该边界在某些情形下是紧的，并重现了已知结果，例如压缩性与遍历性、超混合性以及Doeblin极小化条件之间的联系。将我们的方法特化到$L_p$空间后，相比经典的Riesz-Thorin插值方法取得了显著改进。此外，通过利用Orlicz空间提供的灵活性，我们能够处理平稳分布为重尾分布的情形——这是一个尚未被充分研究的领域。作为本文提出框架的应用，我们引入了更紧的马尔可夫过程混合时间上界、更优的MCMC方法指数浓度界，以及更优的燃烧期下界。最后，我们展示了如何利用这些结果证明一系列马尔可夫随机变量的测度集中现象。