Adaptive invariant density estimation for continuous-time mixing Markov processes under sup-norm risk

Abstract

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 and in particular 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. To contribute to the statistical understanding in more general situations, we demonstrate how combining βmixing assumptions on the process and heat kernel bounds on the transition density representing controls on the longand short-time transitional behaviour, allow to obtain sup-norm and L kernel invariant density estimation rates that match the well-understood case of reversible multidimensional diffusion processes and are faster than in a sampled discrete data scenario. Moreover, we demonstrate how, up to log-terms, optimal sup-norm adaptive invariant density estimation can be achieved within our framework, based on tight uniform moment bounds and deviation inequalities for empirical processes associated to additive functionals of Markov processes. The underlying assumptions are verifiable with classical tools from stability theory of continuous-time Markov processes and PDE techniques, which opens the door to evaluate statistical performance for a vast amount of popular Markov models. We highlight this point by showing how multidimensional jump SDEs with Lévy-driven jump part under different coefficient assumptions can be seamlessly integrated into our framework, thus establishing novel adaptive sup-norm estimation rates for this class of processes. MSC2020 subject classifications: Primary 62M05; secondary 62G05, 62G20, 60G10, 60J25