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：我们针对奥尔利希空间中的马尔可夫过程提出了一种新颖的收敛概念，其推广了通常与$L_p$空间相关的方法。在证明马尔可夫算子在奥尔利希空间中具有收缩性之后，我们的关键技术贡献是给出了其收缩系数的上界，该上界具有闭式表达式。该界在某些设定下是紧的，并且恢复了若干经典结论，例如收缩性与遍历性、超混合性及Doeblin次要化条件之间的联系。将我们的方法特化到$L_p$空间，可对经典的Riesz-Thorin插值方法实现显著改进。此外，通过利用奥尔利希空间所提供的灵活性，我们能够处理平稳分布为重尾分布的情形——这是一个目前研究严重不足的设定。作为本文所提出框架的应用，我们给出了马尔可夫过程混合时间的更紧界、MCMC方法更优的指数浓度界，以及预热期的更优下界。最后，我们展示了如何利用我们的结果来证明马尔可夫随机变量序列的测度集中现象。