Doeblin coefficients are a classical tool for analyzing the ergodicity and exponential convergence rates of Markov chains. Propelled by recent works on contraction coefficients of strong data processing inequalities, we investigate whether Doeblin coefficients also exhibit some of the notable properties of canonical contraction coefficients. In this paper, we present several new structural and geometric properties of Doeblin coefficients. Specifically, we show that Doeblin coefficients form a multi-way divergence, exhibit tensorization, and possess an extremal trace characterization. We then show that they also have extremal coupling and simultaneously maximal coupling characterizations. By leveraging these characterizations, we demonstrate that Doeblin coefficients act as a nice generalization of the well-known total variation (TV) distance to a multi-way divergence, enabling us to measure the "distance" between multiple distributions rather than just two. We then prove that Doeblin coefficients exhibit contraction properties over Bayesian networks similar to other canonical contraction coefficients. We additionally derive some other results and discuss an application of Doeblin coefficients to distribution fusion. Finally, in a complementary vein, we introduce and discuss three new quantities: max-Doeblin coefficient, max-DeGroot distance, and min-DeGroot distance. The max-Doeblin coefficient shares a connection with the concept of maximal leakage in information security; we explore its properties and provide a coupling characterization. On the other hand, the max-DeGroot and min-DeGroot measures extend the concept of DeGroot distance to multiple distributions.

翻译：Doeblin系数是分析马尔可夫链遍历性与指数收敛速率的经典工具。受近期关于强数据处理不等式收缩系数研究的推动，我们探讨Doeblin系数是否也展现出典型收缩系数的某些显著性质。本文提出了Doeblin系数的若干新结构与几何性质。具体而言，我们证明Doeblin系数构成一种多向散度，具有张量化特性及极值迹表征。进而揭示其同时存在极值耦合与同步最大耦合表征。利用这些表征，我们论证Doeblin系数可作为著名全变差（TV）距离向多向散度的自然推广，从而能够度量多个分布（而非仅两个）之间的“距离”。随后证明Doeblin系数在贝叶斯网络上具有与其他典型收缩系数相似的收缩性质。我们还推导了其他相关结果，并讨论了Doeblin系数在分布融合中的应用。最后作为补充，我们引入并探讨了三个新度量：最大Doeblin系数、最大DeGroot距离与最小DeGroot距离。最大Doeblin系数与信息安全中的最大泄漏概念存在关联，我们探究其性质并提供耦合表征。另一方面，最大DeGroot与最小DeGroot度量将DeGroot距离的概念推广至多分布情形。