On Doeblin Curves and Their Properties

Doeblin coefficients are fundamental tools in the analysis of Markov chains for establishing ergodicity and exponential convergence rates. However, strong conditions, such as Doeblin coefficients being bounded away from 0, are typically required to yield useful convergence or information contraction guarantees. Our work aims to illuminate the contraction behavior of Markov kernels under more relaxed conditions, such as the case where Doeblin coefficients are 0. To do this, we introduce the notion of a Doeblin curve—a nonlinear function that quantifies the contraction behavior of a Markov kernel on a collection of input distributions. We develop new variational characterizations of Doeblin coefficients and use them to derive useful bounds on the Doeblin curve. In the course of this analysis, we present several properties of Doeblin curves and define power-constrained Doeblin curves. Furthermore, our analysis motivates a generalized definition of differential privacy in the group setting. We discuss this motivation and several properties of this definition to lay the groundwork for its application in future.