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，我们探究Doeblin系数是否也具有经典收缩系数的显著性质。本文提出了Doeblin系数的若干新结构与几何特性。具体而言，我们证明Doeblin系数构成一种多路散度，具有张量化性质，并拥有极值迹刻画。随后给出其极值耦合与同时最大耦合刻画。利用这些刻画，我们论证Doeblin系数是经典全变差距离到多路散度的优良推广，从而可度量多个分布间的"距离"而非仅限于两个分布。进一步证明Doeblin系数在贝叶斯网络上具有与其他经典收缩系数类似的收缩性质。此外我们推导了若干其他结果，并讨论Doeblin系数在分布融合中的应用。最后作为补充研究，我们引入并讨论三个新量：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露概念存在关联，我们探索其性质并给出耦合刻画。而最大及最小DeGroot度量则将DeGroot距离概念扩展至多分布情形。