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，超越了传统$L_p$空间框架。在证明马尔可夫算子在Orlicz空间具有压缩性后，我们的关键技术贡献是给出了其压缩系数的闭式上界。该界在某些情形下是紧的，并涵盖了已知结果，例如压缩性与遍历性、超混合性及Doeblin小化条件之间的关联。将我们的方法特化为$L_p$空间后，可显著改进经典的Riesz-Thorin插值方法。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这一研究严重不足的情形。作为本文所提框架的应用，我们给出了马尔可夫过程混合时间的更紧界、马尔可夫链蒙特卡洛方法的指数浓度上界改进，以及更优的退火期下界。最后，我们展示了如何利用这些结果证明马尔可夫随机变量序列的测度集中现象。