Mixing it up: A general framework for Markovian statistics beyond reversibility and the minimax paradigm

Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of estimators for the characteristics of the process in the minimax sense, it restricts the applicability of results to a rather constrained set of stochastic processes. In particular, among other drawbacks, it hardly allows incorporating jump structures. As a consequence, for many models of applied and theoretical interest, no statement can be made about the robustness of typical statistical procedures beyond the beautiful, but limited framework available in the literature. This motivates us to go a different route, by not asking what rate-optimal results can be obtained for a specific type of Markov process, but what are the essential stability properties required in general that allow building a rigorous and robust statistical theory upon. We provide an answer to this question by showing that mixing properties are sufficient to obtain deviation and moment bounds for integral functionals of general Markov processes. Together with some unavoidable technical but not structural assumptions on the semigroup, these allow to derive convergence rates for kernel invariant density estimation which are known to be optimal in the much more restrictive context of reversible continuous diffusion processes, thus indicating the potential comprehensiveness of the mixing framework for various statistical purposes. We underline the usefulness of our general modelling idea by establishing new upper bounds on convergence rates of kernel invariant density estimation for scalar L\'evy-driven OU processes and multivariate L\'evy jump diffusions, both for the pointwise $L^2$-risk and the $\sup$-norm risk, by showing how they can be seamlessly integrated into our framework.